{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Linear Equations\n", "The equations in the previous lab included one variable, for which you solved the equation to find its value. Now let's look at equations with multiple variables. For reasons that will become apparent, equations with two variables are known as linear equations.\n", "\n", "## Solving a Linear Equation\n", "Consider the following equation:\n", "\n", "\\begin{equation}2y + 3 = 3x - 1 \\end{equation}\n", "\n", "This equation includes two different variables, **x** and **y**. These variables depend on one another; the value of x is determined in part by the value of y and vice-versa; so we can't solve the equation and find absolute values for both x and y. However, we *can* solve the equation for one of the variables and obtain a result that describes a relative relationship between the variables.\n", "\n", "For example, let's solve this equation for y. First, we'll get rid of the constant on the right by adding 1 to both sides:\n", "\n", "\\begin{equation}2y + 4 = 3x \\end{equation}\n", "\n", "Then we'll use the same technique to move the constant on the left to the right to isolate the y term by subtracting 4 from both sides:\n", "\n", "\\begin{equation}2y = 3x - 4 \\end{equation}\n", "\n", "Now we can deal with the coefficient for y by dividing both sides by 2:\n", "\n", "\\begin{equation}y = \\frac{3x - 4}{2} \\end{equation}\n", "\n", "Our equation is now solved. We've isolated **y** and defined it as 3x-4/2\n", "\n", "While we can't express **y** as a particular value, we can calculate it for any value of **x**. For example, if **x** has a value of 6, then **y** can be calculated as:\n", "\n", "\\begin{equation}y = \\frac{3\\cdot6 - 4}{2} \\end{equation}\n", "\n", "This gives the result 14/2 which can be simplified to 7.\n", "\n", "You can view the values of **y** for a range of **x** values by applying the equation to them using the following Python code:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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xy
0-10-17.0
1-9-15.5
2-8-14.0
3-7-12.5
4-6-11.0
5-5-9.5
6-4-8.0
7-3-6.5
8-2-5.0
9-1-3.5
100-2.0
111-0.5
1221.0
1332.5
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1555.5
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" ], "text/plain": [ " x y\n", "0 -10 -17.0\n", "1 -9 -15.5\n", "2 -8 -14.0\n", "3 -7 -12.5\n", "4 -6 -11.0\n", "5 -5 -9.5\n", "6 -4 -8.0\n", "7 -3 -6.5\n", "8 -2 -5.0\n", "9 -1 -3.5\n", "10 0 -2.0\n", "11 1 -0.5\n", "12 2 1.0\n", "13 3 2.5\n", "14 4 4.0\n", "15 5 5.5\n", "16 6 7.0\n", "17 7 8.5\n", "18 8 10.0\n", "19 9 11.5\n", "20 10 13.0" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "# Create a dataframe with an x column containing values from -10 to 10\n", "df = pd.DataFrame ({'x': range(-10, 11)})\n", "\n", "# Add a y column by applying the solved equation to x\n", "# solving the equation (3x - 4)/2 = y\n", "df['y'] = (3*df['x'] - 4) / 2\n", "\n", "#Display the dataframe\n", "df" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also plot these values to visualize the relationship between x and y as a line. For this reason, equations that describe a relative relationship between two variables are known as *linear equations*:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib import pyplot as plt\n", "\n", "plt.plot(df.x, df.y, color=\"grey\", marker = \"o\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In a linear equation, a valid solution is described by an ordered pair of x and y values. For example, valid solutions to the linear equation above include:\n", "- (-10, -17)\n", "- (0, -2)\n", "- (9, 11.5)\n", "\n", "The cool thing about linear equations is that we can plot the points for some specific ordered pair solutions to create the line, and then interpolate the x value for any y value (or vice-versa) along the line." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Intercepts\n", "When we use a linear equation to plot a line, we can easily see where the line intersects the X and Y axes of the plot. These points are known as *intercepts*. The *x-intercept* is where the line intersects the X (horizontal) axis, and the *y-intercept* is where the line intersects the Y (horizontal) axis.\n", "\n", "Let's take a look at the line from our linear equation with the X and Y axis shown through the origin (0,0)." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(df.x, df.y, color=\"grey\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.grid()\n", "\n", "## add axis lines for 0,0\n", "plt.axhline()\n", "plt.axvline()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The x-intercept is the point where the line crosses the X axis, and at this point, the **y** value is always 0. Similarly, the y-intercept is where the line crosses the Y axis, at which point the **x** value is 0. So to find the intercepts, we need to solve the equation for **x** when **y** is 0.\n", "\n", "For the x-intercept, our equation looks like this:\n", "\n", "\\begin{equation}0 = \\frac{3x - 4}{2} \\end{equation}\n", "\n", "Which can be reversed to make it look more familar with the x expression on the left:\n", "\n", "\\begin{equation}\\frac{3x - 4}{2} = 0 \\end{equation}\n", "\n", "We can multiply both sides by 2 to get rid of the fraction:\n", "\n", "\\begin{equation}3x - 4 = 0 \\end{equation}\n", "\n", "Then we can add 4 to both sides to get rid of the constant on the left:\n", "\n", "\\begin{equation}3x = 4 \\end{equation}\n", "\n", "And finally we can divide both sides by 3 to get the value for x:\n", "\n", "\\begin{equation}x = \\frac{4}{3} \\end{equation}\n", "\n", "Which simplifies to:\n", "\n", "\\begin{equation}x = 1\\frac{1}{3} \\end{equation}\n", "\n", "So the x-intercept is 11/3 (approximately 1.333).\n", "\n", "To get the y-intercept, we solve the equation for y when x is 0:\n", "\n", "\\begin{equation}y = \\frac{3\\cdot0 - 4}{2} \\end{equation}\n", "\n", "Since 3 x 0 is 0, this can be simplified to:\n", "\n", "\\begin{equation}y = \\frac{-4}{2} \\end{equation}\n", "\n", "-4 divided by 2 is -2, so:\n", "\n", "\\begin{equation}y = -2 \\end{equation}\n", "\n", "This gives us our y-intercept, so we can plot both intercepts on the graph:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(df.x, df.y, color=\"grey\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.grid()\n", "\n", "## add axis lines for 0,0\n", "plt.axhline()\n", "plt.axvline()\n", "plt.annotate('x-intercept',(1.333, 0))\n", "plt.annotate('y-intercept',(0,-2))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The ability to calculate the intercepts for a linear equation is useful, because you can calculate only these two points and then draw a straight line through them to create the entire line for the equation." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Slope\n", "It's clear from the graph that the line from our linear equation describes a slope in which values increase as we travel up and to the right along the line. It can be useful to quantify the slope in terms of how much **x** increases (or decreases) for a given change in **y**. In the notation for this, we use the greek letter Δ (*delta*) to represent change: ( Δx = x-intercept)\n", "\n", "\\begin{equation}slope = \\frac{\\Delta{y}}{\\Delta{x}} \\end{equation}\n", "\n", "Sometimes slope is represented by the variable ***m***, and the equation is written as:\n", "\n", "\\begin{equation}m = \\frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\end{equation}\n", "\n", "Although this form of the equation is a little more verbose, it gives us a clue as to how we calculate slope. What we need is any two ordered pairs of x,y values for the line - for example, we know that our line passes through the following two points:\n", "- (0,-2)\n", "- (6,7)\n", "\n", "We can take the x and y values from the first pair, and label them x1 and y1; and then take the x and y values from the second point and label them x2 and y2. Then we can plug those into our slope equation:\n", "\n", "\\begin{equation}m = \\frac{7 - -2}{6 - 0} \\end{equation}\n", "\n", "This is the same as:\n", "\n", "\\begin{equation}m = \\frac{7 + 2}{6 - 0} \\end{equation}\n", "\n", "That gives us the result 9/6 which is 11/2 or 1.5 .\n", "\n", "So what does that actually mean? Well, it tells us that for every change of **1** in x, **y** changes by 11/2 or 1.5. So if we start from any point on the line and move one unit to the right (along the X axis), we'll need to move 1.5 units up (along the Y axis) to get back to the line.\n", "\n", "You can plot the slope onto the original line with the following Python code to verify it fits:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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Tp4/VsSRAqfiIiEhQ8vl8LF++nPXr1wPQvn17nE4nLVu2tDiZBDIVHxERCTqlpaW4XC6Ki4sBGDx4MGPHjtWhLbkp/YSIiEhQ2bdvHwsXLqS6upqYmBimTZvGHXfcYXUsCRIqPiIiEhS8Xi/Hjx9n+/btANx22204nU5SUlIszSXBRcVHREQC3oULF8jLy+Ps2bMAZGZmMnr0aMLDwy1OJsFGxUdERALa7t27mTdvHm63m/DwcLKzs+ndu7fVsSRIqfiIiEhA8nq9LFmyhC1btgDQoUMHkpKS6N69u8XJJJiFWR2gMb388ss4HI6rPnr16mV1LBERaaBz587x4Ycf1paeoUOH8t3vfpeoqCiLk0mws90enz59+rBs2bLay3pqo4hIcNm5cycLFiygpqaGuLg4ZsyYQbdu3fB4PFZHExuwXSuIiIigXbt2VscQEZEG8ng8LF68mM8//xyATp06kZ2dTVJSksXJxE5sV3z2799PWloaMTExZGZm8sYbb9CxY8cbLu92u3G73bWXy8vLgcsPwMb87+LKuuz6H4vd5wP7z2j/+bxXfW7HOYN5G549e5Y5c+Zw5swZ4PKhrfvuu4+wsLBr5grG+erL7jM25Xz1XafDGGMa/atbZNGiRVy8eJGePXty8uRJXnnlFYqLi/nyyy9JTEy87n1efvllXnnllWuu/8tf/kJcXFxTRxaRZuL2wS83Xf5f763BXqL1LOiAcf78eY4fP47f7yciIoJOnTrd8He2yI1UVlby0EMPUVZWVudeQlsVn28qLS2lU6dOvPPOOzz++OPXXeZ6e3zS09M5e/Zso+5e9Xg8FBUVMXbsWCIjIxttvYHC7vOB/We0+3yVNV76v7YCgC3PDyc5PsbiRI0v2LZhTU0NS5cu5YsvvgAuH9qaPn06CQkJ110+2Oa7FXafsSnnKy8vp3Xr1jctPrY71PV1KSkp9OjRgwMHDtxwmejoaKKjo6+5PjIyskl+6JpqvYHC7vOB/We063yRxvGPzyMjbDnjFcGwDU+fPo3L5eLMmTM4HA5GjBhRe2jrZoJhvm/L7jM2xXz1XZ+ti8/Fixc5ePAgDz/8sNVRREQEMMawbds2Fi1ahNfrJSEhgZycHDp37mx1NAkRtio+P//5z5k6dSqdOnXixIkTvPTSS4SHh/Pggw9aHU1EJOS53W4WLlzIzp07Abj99tuZMWMG8fHxFieTUGKr4nP8+HEefPBBzp07R2pqKsOGDWPDhg2kpqZaHU1EJKSdOnUKl8vFuXPncDgcjBo1iqFDh+JwOG5+Z5FGZKvi89e//tXqCCIi8jXGGLZu3crixYvx+XwkJSWRk5NT58uMiDQlWxUfEREJHG63m/nz57Nr1y4AunfvTlZWll4qRCyl4iMiIo3u5MmT5OXlceHCBcLCwhg9ejSZmZk6tCWWU/EREZFGY4xh8+bNLF26FJ/PR3JyMk6nkw4dOlgdTQRQ8RERkUZSXV3NvHnz2LNnDwA9e/Zk+vTpxMbGWpxM5B9UfERE5FsrLi7G5XJRWlpKWFgYY8eOJSMjQ4e2JOCo+IiIyC0zxrBhwwaWLVuG3+8nJSUFp9PJbbfdZnU0ketS8RERkVtSVVVFQUEBX331FQC9e/dm6tSpxMTY733QxD5UfEREpMGOHTuGy+WivLyc8PBwxo8fz913361DWxLwVHxERKTejDGsW7eO5cuXY4yhZcuW5Obm0q5dO6ujidSLio+IiNTLpUuXKCgo4MCBAwD07duXKVOmEB0dbXEykfpT8RERkZs6cuQI+fn5VFRUEBERwYQJExg0aJAObUnQUfEREZEb8vv9rFmzhk8//RRjDK1bt8bpdNK2bVuro4ncEhUfERG5rosXLzJnzhz+/ve/A9C/f38mTZpEVFSUxclEbp2Kj4iIXOPQoUPMnj2bixcvEhkZyaRJkxgwYIDVsUS+NRUfERGp5ff7WbVqFatXrwYgNTWV3NxcUlNTLU4m0jhUfEREBICKigpmz57N4cOHARg4cCATJ04kMjLS2mAijUjFR0REOHjwILNnz6ayspLIyEimTJnCnXfeaXUskUan4iMiEsL8fj8rV65kzZo1ALRt2xan00nr1q0tTibSNFR8RERCVHl5Ofn5+Rw9ehSAu+66i/Hjx+vQltiaio+ISAjav38/c+bMoaqqiqioKKZOnUrfvn2tjiXS5FR8RERCiM/nY/ny5axfvx6A9u3b43Q6admypcXJRJqHio+ISIgoLS0lPz+f48ePAzB48GDGjh1LRIT+FEjo0E+7iEgI2Lt3L3PnzqW6upro6GimT5/OHXfcYXUskWan4iMiYmM+n4+ioiI2btwIQFpaGk6nkxYtWlicTMQaKj4iIjZ14cIFXC4XJ06cAODee+9lzJgxhIeHW5xMxDoqPiIiNrR7927mzZuH2+0mJiaGrKwsevbsaXUsEcup+IiI2Ijf72fx4sV8/vnnAHTo0AGn00lycrLFyUQCg4qPiIhNnD9/nv3791NVVQXA0KFDGTlypA5tiXyNio+IiA18+eWXzJ8/n5qaGmJjY5kxYwbdu3e3OpZIwFHxEREJYh6P56pDW/Hx8Tz22GN6QUKRG1DxEREJUmfPniUvL4/Tp08Dlw9tXbx4kcTERIuTiQQuFR8RkSC0Y8cOFi5ciMfjIT4+nhkzZtCxY0cKCwutjiYS0FR8RESCSE1NDYsWLWL79u0AdOnShRkzZpCYmIjH47E2nEgQUPEREQkSp0+fxuVycebMGRwOByNGjOC+++4jLCzM6mgiQUPFR0QkwBlj2L59O4WFhXi9XhISEsjJyaFz585WRxMJOio+IiIBrKamhoULF/LFF18AcPvttzNjxgzi4+MtTiYSnFR8REQCVElJCXl5eZw7dw6Hw8HIkSMZNmwYDofD6mgiQUvFR0QkwBhj2Lp1K4sXL8bn85GYmIjT6aRjx45WRxMJeio+IiIBxO12M3/+fHbt2gVA9+7dycrKIi4uzuJkIvag4iMiEiBOnjyJy+Xi/PnzhIWFMXr0aDIzM3VoS6QRqfiIiFjMGMPmzZtZunQpPp+P5ORknE4nHTp0sDqaiO2o+IiIWKi6upp58+axZ88eAHr27Mn06dOJjY21OJmIPan4iIhYpLi4GJfLRWlpKWFhYYwdO5aMjAwd2hJpQio+IiLNzBjDhg0bWLZsGX6/n5SUFJxOJ7fddpvV0URsT8VHRKQZVVVVMXfuXPbt2wdA7969mTp1KjExMRYnEwkNtnyDl/fff5/OnTsTExNDRkYGmzZtsjqSiAjHjh3jgw8+YN++fYSHhzNp0iScTqdKj0gzst0en7/97W/MnDmTDz74gIyMDN577z3Gjx/Pvn37aNOmjdXxRCQEGWNYt24dy5cvxxhDy5YtcTqdtG/f3upoIiHHdsXnnXfe4YknnuDRRx8F4IMPPmDhwoX88Y9/5Pnnn6/3eiprvETUeBstl8fjxe27vN5IY78TF+0+H9h/RrvPV/m1x3NljY/IyMZ7fNfl0qVKFiyYz8GDfwcc9O7dh4kTJxIdHXVVpsZg921o9/nA/jM25Xz1fTw5jDGmUb+yhWpqaoiLi8PlcpGVlVV7/SOPPEJpaSlz58695j5utxu32117uby8nPT0dNKf+YSwaL1SqoiISDDwuys59t4/UVZWRlJS0g2Xs9U5PmfPnsXn89G2bdurrm/bti2nTp267n3eeOMNkpOTaz/S09ObI6qI2EiEz8vT6/5GSlW51VFE5CZsd6iroV544QVmzpxZe/nKHp+1vxxeZ2NsKI/Hy4oVKxg1ahSRkfb7ttt9PrD/jHafr7LGR+ZvVgHw2cyhJMVHN8p6HceOEffoI0Rs2MAzCWc588ePKVy8iMOHjwCXn7U1duxYoqIiG+Xr1cXu29Du84H9Z2zK+crLy+n43s2Xs9V3tXXr1oSHh1NSUnLV9SUlJbRr1+6694mOjiY6+tpfgMnxsSTFN94rp3o8HqLDITk+hsjIpv8F2NzsPh/Yf0a7z/f1c3qS4qNJbozH9/z58MgjcOHC5a+xaBFfPfEYxXffTWxkBJMnT2bAgAHf/uvUk923od3nA/vP2JTzOXyeei1nq0NdUVFR3HXXXSxfvrz2Or/fz/Lly8nMzLQwmYjYSk0NPPccTJtWW3quuK+wkL4XL/Lkk082a+kRkfqxVfEBmDlzJv/xH//Bxx9/zJ49e3jqqae4dOlS7bO8RES+lUOH4L774J13rntzuN9Pdl4eqRG22qEuYhu2e2Q+8MADnDlzhhdffJFTp04xYMAAFi9efM0JzyIiDTZ7Njz2GJSV1bmY48gReOop+OtfmymYiNSX7YoPwE9+8hN+8pOfWB1DROzC7Yaf/xz+z/+p3/Ldu0MDXjdMRJqPLYuPiEijOXAAHngAPv+8fss/9BB88AEkJjZtLhG5JbY7x0dEpNF88gkMGlS/0hMTA//f/wd/+pNKj0gA0x4fEZFvqqqCmTMv77mpjzvuuFyS+vZt2lwi8q2p+IiIfN2+ffBP/wRffFG/5R95BN5/H+LjmzaXiDQKHeoSEbniz3+Gu+6qX+mJi4NZsy5/qPSIBA3t8RERqayEn/4UPvywfsv37Xv50NYddzRtLhFpdNrjIyKhbfduGDy4/qXnhz+EjRtVekSClIqPiISuWbPgnntg166bL5uQcPlQ2H/8x+XDXCISlHSoS0RCz8WL8M8/hv/8z/ot37//5UNbPXo0bS4RaXLa4yMiIaXnmcMkDL+v/qXnn/8ZNmxQ6RGxCe3xEZHQYAzf2b6Yl5f/gXBvzc2XT0q6/IKEublNn01Emo2Kj4jYX0UFUU88yZtL6vmmoXfdBX/7G9x+e9PmEpFmp0NdImJv27bBoEFE/K2epednP4O1a1V6RGyqwcXnkUceYfXq1U2RRUSk8RgD//f/Qmbm5TcavZmUFJgzB957D6KjmzqdiFikwcWnrKyMMWPG0L17d15//XWKi4ubIpeIyLdz7Bj84hfgdt982YyMy3uGsrKaPJaIWKvBxaegoIDi4mKeeuop/va3v9G5c2cmTpyIy+XC4/E0RUYRkYbr2PHye2jdzHPPwerV0Llzk0cSEevd0jk+qampzJw5kx07drBx40a6devGww8/TFpaGs8++yz79+9v7JwiIg1ijGFb//58MWDA9Rdo2RLmz4e334aoqGbNJiLW+VYnN588eZKioiKKiooIDw9n0qRJ7Ny5k969e/Puu+82VkYRkQapqamhoKCAefPmsWDiRMrS0q663ZuZCdu3w5Qp1gQUEcs0uPh4PB7y8/OZMmUKnTp1Ii8vj2eeeYYTJ07w8ccfs2zZMj755BNeffXVpsgrIlKnkpIS/vCHP/DFF1/gcDi4b+JEkhYtwsTEAPD+vblcKlwM6ekWJxURKzT4dXzat2+P3+/nwQcfZNOmTQy4zm7kkSNHkpKS0gjxRETqxxjD559/zuLFi/F6vSQmJpKTk0OnTp0AcP/uA54oPMLqrnfxvchIi9OKiFUaXHzeffddcnNzifl//z1dT0pKCocOHfpWwURE6svtdrNgwQK+/PJLALp3705WVhZxX3szUd9D32X13iVWRRSRANHg4vPwww83RQ4RkVty8uRJXC4X58+fJywsjFGjRjFkyBAcDofV0UQkAOktK0QkKBlj2Lx5M0uXLsXn85GcnExOTg7pOndHROqg4iMiQae6upr58+eze/duAHr27Mn06dOJjY21OJmIBDoVHxEJKsXFxbhcLkpLSwkLC2Ps2LFkZGTo0JaI1IuKj4gEBWMMGzdupKioCL/fT0pKCk6nk9tuu83qaCISRFR8RCTgVVVVMXfuXPbt2wfAHXfcwbRp0+p8dqmIyPWo+IhIQDt27Bj5+fmUlZURHh7OuHHjuOeee3RoS0RuiYqPiAQkYwzr1q1jxYoV+P1+WrZsidPppH379lZHE5EgpuIjIgGnsrKSgoKC2jc87tu3L1OmTCE6OtriZCIS7FR8RCSgHD16lPz8fMrLy4mIiGDChAkMGjRIh7ZEpFGo+IhIQDDGsGbNGlauXIkxhlatWpGbm0vbtm2tjiYiNqLiIyKWu3TpEnPmzOHgwYMA3HnnnUyePJmoqCiLk4mI3aj4iIilDh8+TH5+PhcvXiQiIoJJkyYxYMAAHdoSkSah4iMilvD7/axevZrVq1djjCE1NZXc3FxSU1OtjiYiNqbiIyLNrqKigjlz5nDo0CEABgwYwKRJk4iMjLQ4mYjYnYqPiDSrgwcPMmfOHC5dukRkZCRTpkzhzjvvtDqWiIQIFR8RaRZ+v59PP/2Uzz77DIC2bdvidDpp3bq1xclEJJSo+IhIkysvLyc/P5+jR48CcNdddzF+/Hgd2hKRZqfiIyJNav/+/RQUFFBZWUlUVBRTp06lb9++VscSkRCl4iMiTcLn87FixQrWrVsHQLt27XA6nbRq1criZCISylR8RKTRlZWVkZ+fz7FjxwC45557GDduHBER+pUjItbSbyERaVT79u2joKCA6upqoqOjmTZtGr1797Y6logIoOIjIo3E5/OxbNkyNmzYAEBaWhpOp5MWLVpYnExE5B9UfETkW7tw4QL5+fkUFxcDkJGRwdixYwkPD7c4mYjI1cKsDtCYOnfujMPhuOrjzTfftDqWiK3t2bOH3//+9xQXFxMTE8MDDzzAhAkTVHpEJCDZbo/Pq6++yhNPPFF7OTEx0cI0Ivbl9XpZunQpmzdvBqBDhw7k5OSQkpJibTARkTrYrvgkJibSrl07q2OI2Nr58+dxuVycPHkSgCFDhjBq1Cjt5RGRgGe74vPmm2/y2muv0bFjRx566CGeffbZOp9C63a7cbvdtZfLy8sB8Hg8eDyeRst1ZV2Nuc5AYvf5wP4z1ne+3bt3U1hYSE1NDbGxsUydOpVu3brh9/vx+/3NEfWWeDzeqz6343bUz2jws/uMTTlffdfpMMaYRv/qFnnnnXcYNGgQLVu2ZN26dbzwwgs8+uijvPPOOze8z8svv8wrr7xyzfV/+ctfiIuLa8q4IkHF7/dTXFzMuXPnAIiPj6dTp05ERUVZnKx+3D745abL/wS9NdhLtHZOidhKZWUlDz30EGVlZSQlJd1wuYAvPs8//zy/+c1v6lxmz5499OrV65rr//jHP/KjH/2IixcvEh0dfd37Xm+PT3p6OmfPnq3zG9dQHo+HoqIixo4da8v3J7L7fGD/Geua79y5c8yZM4fTp08Dlw9tDR8+nLCw4Hl+RGWNl/6vrQBgy/PDSY6PsThR4wvln1G7sPuMTTlfeXk5rVu3vmnxCfhDXc899xw/+MEP6lyma9eu170+IyMDr9fL4cOH6dmz53WXiY6Ovm4pioyMbJIfuqZab6Cw+3xg/xm/Od8XX3zBggUL8Hg8xMXFkZ2dze23325hwlsTaRz/+DwyIqS2od3YfT6w/4xNMV991xfwxSc1NZXU1NRbuu/27dsJCwujTZs2jZxKxP48Hg+FhYVs374duPxyEdnZ2XqmpIgEtYAvPvW1fv16Nm7cyMiRI0lMTGT9+vU8++yzfO9739Mrx4o00JkzZ8jLy+PMmTMAjBgxIugObYmIXI9tik90dDR//etfefnll3G73XTp0oVnn32WmTNnWh1NJGgYY9ixYwdLlizB6/WSkJBAdnY2Xbp0sTqaiEijsE3xGTRoUO17BIlIw9XU1HD06FF27NgBXD53bsaMGSQkJFicTESk8dim+IjIrSspKSEvL48LFy7gcDi4//77ue+++3A4HDe/s4hIEFHxEQlhxhg+//xzFi9ejNfrJTIykgceeCAon7UlIlIfKj4iIcrtdrNgwQK+/PJLAG6//XZiY2Pp2LGjxclERJqOio9ICDp58iQul4vz58/jcDgYPXo099xzD4sWLbI6mohIk1LxEQkhxhg2b97M0qVL8fl8JCcnk5OTQ3p6um3fG0hE5OtUfERCRHV1NfPnz2f37t0A9OzZk+nTpxMbG2txMhGR5qPiIxICiouLcblclJaWEhYWxtixY8nIyNCztkQk5Kj4iNiYMYaNGzdSVFSE3+8nJSUFp9PJbbfdZnU0ERFLqPiI2FRVVRVz585l3759ANxxxx1MmzaNmBj7vSu5iEh9qfiI2NCxY8fIz8+nrKyM8PBwxo0bxz333KNDWyIS8lR8RGzEGMO6detYsWIFfr+fFi1akJubS/v27a2OJiISEFR8RGyisrKSgoIC9u/fD0CfPn2YOnUq0dHRFicTEQkcKj4iNnDkyBHy8/OpqKggPDyciRMnMmjQIB3aEhH5BhUfkSBmjGHNmjWsXLkSYwytWrUiNzeXtm3bWh1NRCQgqfiIBKlLly4xZ84cDh48CMCdd97J5MmTiYqKsjiZiEjgUvERCUKHDx8mPz+fixcvEhERwaRJkxgwYIAObYmI3ISKj0gQ8fv9rF69mtWrV2OMITU1FafTSZs2bayOJiISFFR8RIJERUUFc+bM4dChQwAMGDCAiRMn6tCWiEgDqPiIBIGDBw8yZ84cLl26RGRkJJMnT6Z///5WxxIRCToqPiIBzO/38+mnn/LZZ58B0KZNG3Jzc2ndurXFyUREgpOKj0iAKi8vJz8/n6NHjwIwaNAgJkyYQGRkpMXJRESCl4qPSADav38/BQUFVFZWEhUVxdSpU+nbt6/VsUREgp6Kj0gA8fl8rFixgnXr1gHQrl07nE4nrVq1sjiZiIg9qPiIBIiysjLy8/M5duwYAPfccw/jxo0jIkIPUxGRxqLfqCIBYN++fRQUFFBdXU10dDTTpk2jd+/eVscSEbEdFR8RC/l8PpYtW8aGDRsASEtLw+l00qJFC4uTiYjYk4qPiEUuXLhAfn4+xcXFAGRkZDBmzBgd2hIRaUL6DStigT179jB37lzcbjcxMTFMnz6dXr16WR1LRMT2VHxEmpHX62Xp0qVs3rwZgA4dOpCTk0NKSoq1wUREQoSKj0gzOX/+PC6Xi5MnTwIwZMgQRo0aRXh4uMXJRERCh4qPSDPYtWsX8+bNo6amhtjYWLKysujRo4fVsUREQo6Kj0gT8ng8LFmyhK1btwLQsWNHcnJySEpKsjiZiEhoUvERaSJnz57F5XJRUlICwLBhwxg5ciRhYWEWJxMRCV0qPiJN4IsvvmDBggV4PB7i4uKYMWMG3bp1szqWiEjIU/ERaUQej4dFixaxbds2ADp37kx2djaJiYkWJxMREVDxEWk0Z86cIS8vjzNnzgAwfPhwRowYoUNbIiIBRMVHpBFs376dwsJCPB4PCQkJZGdn06VLF6tjiYjIN6j4iHwLNTU1FBYWsmPHDgC6du3KjBkzSEhIsDiZiIhcj4qPyC0qKSnB5XJx9uxZHA4H999/P8OGDdOhLRGRAKbiI9JAxhi2bt3K4sWL8Xq9JCYmkpOTQ6dOnayOJiIiN6HiI9IAPp+PuXPnsnv3bgC6detGVlYW8fHxFicTEZH6UPERqadTp06xb98+ampqcDgcjBo1iqFDh+JwOKyOJiIi9aTiI3ITxhi2bNnCkiVL8Pl8JCUl4XQ6SU9PtzqaiIg0kIqPSB2qq6uZP39+7aGtpKQkHnvsMZKTky1OJiIit0LFR+QGTpw4QV5eHqWlpYSFhTFy5EjOnj1LXFyc1dFEROQWBc3zbn/9618zZMgQ4uLiSElJue4yR48eZfLkycTFxdGmTRt+8Ytf4PV6mzeoBD1jDBs2bODDDz+ktLSUlJQUHnvsMTIyMnQ+j4hIkAuaPT41NTXk5uaSmZnJhx9+eM3tPp+PyZMn065dO9atW8fJkyf5/ve/T2RkJK+//roFiSUYVVVVMW/ePPbu3QtAr169mD59OjExMXg8HovTiYjItxU0xeeVV14BYNasWde9fenSpezevZtly5bRtm1bBgwYwGuvvcavfvUrXn75ZaKiopoxrQSj48eP43K5KCsrIzw8nHHjxnHPPfdoL4+IiI0ETfG5mfXr19OvXz/atm1be9348eN56qmn2LVrFwMHDrzu/dxuN263u/ZyeXk5cPldthvzP/wr67LrXoNgns8Yw8aNG/n000/x+/20aNGCrKws2rdvf9Wh0mCesT7sP5/3qs/tOKf9t6G95wP7z9iU89V3nbYpPqdOnbqq9AC1l0+dOnXD+73xxhu1e5O+bunSpU1yEmtRUVGjrzOQBNt8Xq+Xo0eP1hbelJQUOnTowLZt29i2bdt17xNsMzaUXedz++DKr7wVK1YQHW5pnCZl1214hd3nA/vP2BTzVVZW1ms5S4vP888/z29+85s6l9mzZw+9evVqsgwvvPACM2fOrL1cXl5Oeno648aNIykpqdG+jsfjoaioiLFjxxIZGdlo6w0UwTjfsWPHKCgooKKigvDwcMaOHcvAgQNveGgrGGdsCLvPV1nj5ZebVgAwatQokuNjLE7U+Oy+De0+H9h/xqac78o/sDdjafF57rnn+MEPflDnMl27dq3Xutq1a8emTZuuuq6kpKT2thuJjo4mOjr6musjIyOb5IeuqdYbKIJhPmMMa9asYeXKlRhjaNWqFU6ns86fk68Lhhm/DbvOF2n+UWgjIyNsOeMVdt2GV9h9PrD/jE0xX33XZ2nxSU1NJTU1tVHWlZmZya9//WtOnz5NmzZtgMu70pKSkujdu3ejfA0JfpcuXWLOnDkcPHgQgH79+jF58uTrll8REbGfoDnH5+jRo5w/f56jR4/i8/nYvn07cPlNIhMSEhg3bhy9e/fm4Ycf5q233uLUqVP8y7/8C08//bT+qAkAhw8fJj8/n4sXLxIREcHEiRPrPLQlIiL2EzTF58UXX+Tjjz+uvXzlWVorV67k/vvvJzw8nAULFvDUU0+RmZlJfHw8jzzyCK+++qpVkSVA+P1+PvvsM1atWoUxhtatW5Obm1u7Z1BEREJH0BSfWbNm3fA1fK7o1KkThYWFzRNIgsLFixeZPXs2hw4dAmDAgAFMnDhRr+skIhKigqb4iDTU3//+d2bPns2lS5eIjIxk8uTJ9O/f3+pYIiJiIRUfsR2/38+qVatYvXo1AG3atCE3N5fWrVtbnExERKym4iO2UlFRQX5+PkeOHAFg0KBBTJgwwdZPCxURkfpT8RHbOHDgAHPmzKGyspKoqCimTJlCv379rI4lIiIBRMVHgp7f72fFihWsXbsWuPyClU6nk1atWlmcTEREAo2KjwS1srIy8vPzOXbsGAB3330348ePJyJCP9oiInIt/XWQoPXVV19RUFBAVVUV0dHRTJ06lT59+lgdS0REApiKjwQdn8/HsmXL2LBhAwBpaWk4nU5atGhhcTIREQl0Kj4SVEpLS3G5XBQXFwOQkZHBmDFjdGhLRETqRX8tJGjs3buXuXPnUl1dTUxMDNOnT6dXr15WxxIRkSCi4iMBz+v1UlRUxKZNmwC47bbbcDqdpKSkWBtMRESCjoqPBLTz58/jcrk4efIkAJmZmYwePZrw8HCLk4mISDBS8ZGAtWvXLubPn4/b7SY2NpasrCx69OhhdSwREQliKj4ScLxeL0uWLGHLli0ApKenk5OTQ3JyssXJREQk2Kn4SEA5d+4ceXl5lJSUADBs2DBGjhxJWFiYxclERMQOVHwkYOzcuZMFCxZQU1NDXFwcM2bMoFu3blbHEhERG1HxEct5PB4WLVrEtm3bAOjcuTPZ2dkkJiZanExEROxGxUcsdebMGVwuF6dPnwZg+PDhjBgxQoe2RESkSaj4iGW2b99OYWEhHo+H+Ph4srOz6dq1q9WxRETExlR8pNnV1NRQWFjIjh07AOjSpQvZ2dkkJCRYnExEROxOxUeaVUlJCS6Xi7Nnz+JwOLj//vsZNmyYDm2JiEizUPGRZmGMYdu2bSxatAiv10tCQgI5OTl07tzZ6mgiIhJCVHykybndbhYuXMjOnTsB6NatG1lZWcTHx1ucTEREQo2KjzSpU6dOkZeXx/nz53E4HIwaNYqhQ4ficDisjiYiIiFIxUeahDGGLVu2sGTJEnw+H0lJSeTk5NCxY0ero4mISAhT8ZFGV11dzfz589m9ezcAPXr0YPr06cTFxVmcTEREQp2KjzSqkydPUlBQwIULFwgLC2PMmDHce++9OrQlIiIBQcVHGoUxhjNnzvDxxx/j9/tJTk7G6XTSoUMHq6OJiIjUUvGRb62qqoqCggKKi4sB6NWrF9OmTSM2NtbiZCIiIldT8ZFv5fjx47hcLsrKynA4HIwZM4bMzEwd2hIRkYCk4iO3xBjD+vXrWb58OX6/n5SUFNq0acM999yj0iMiIgFLxUcarLKykrlz5/LVV18B0Lt3byZMmMCKFSssTiYiIlI3FR9pkKNHj5Kfn095eTnh4eFMmDCBu+66C6/Xa3U0ERGRm1LxkXoxxrB27VpWrFiBMYaWLVuSm5tLu3btrI4mIiJSbyo+clOXLl2ioKCAAwcOANCvXz8mT55MdHS0xclEREQaRsVH6nTkyBHy8/OpqKggIiKCiRMnMnDgQJ3ALCIiQUnFR67L7/ezZs0aPv30U4wxtG7dmtzcXNq0aWN1NBERkVum4iPXuHjxIrNnz+bQoUMA9O/fn0mTJhEVFWVxMhERkW9HxUeu8ve//53Zs2dz6dIlIiMjmTRpEgMGDLA6loiISKNQ8RHg8qGtVatWsXr1agDatGmD0+kkNTXV4mQiIiKNR8VHqKioID8/nyNHjgAwcOBAJk6cSGRkpMXJREREGpeKT4g7cOAAc+bMobKykqioKKZMmUK/fv2sjiUiItIkVHxClN/vZ8WKFaxduxaAtm3bkpubS6tWrSxOJiIi0nRUfEJQWVkZ+fn5HDt2DIC7776b8ePHExGhHwcREbE3/aULMV999RUFBQVUVVURHR3N1KlT6dOnj9WxREREmkWY1QHq69e//jVDhgwhLi6OlJSU6y7jcDiu+fjrX//avEEDlM/nY+nSpfz3f/83VVVVtG/fnieffFKlR0REQkrQ7PGpqakhNzeXzMxMPvzwwxsu99FHHzFhwoTayzcqSaGktLQUl8tFcXExAIMHD2bs2LE6tCUiIiEnaP7yvfLKKwDMmjWrzuVSUlL0juFfs3fvXubOnUt1dTUxMTFMnz6dXr16WR1LRETEEkFTfOrr6aef5oc//CFdu3blxz/+MY8++midb6jpdrtxu921l8vLywHweDx4PJ5Gy3VlXY25zrr4fD5WrFjB5s2bAUhLSyMrK4uUlJQmydDc81nB7jPafz7vVZ/bcU77b0N7zwf2n7Ep56vvOh3GGNPoX70JzZo1i2eeeYbS0tJrbnvttdcYNWoUcXFxLF26lJdeeom33nqLn/70pzdc38svv1y7N+nr/vKXvxAXF9eY0ZuN2+3m8OHDVFVVAZCamkr79u0JCwuaU7pEGp3bB7/cdPl/vbcGe4kOtziQiDSqyspKHnroIcrKykhKSrrhcpYWn+eff57f/OY3dS6zZ8+eqw7N1FV8vunFF1/ko48+qn3a9vVcb49Peno6Z8+erfMb11Aej4eioiLGjh3bpK+IvGfPHgoLC3G73cTGxjJlyhS6d+/eZF/viuaaz0p2n9Hu81XWeOn/2goAtjw/nOT4GIsTNT67b0O7zwf2n7Ep5ysvL6d169Y3LT6WHup67rnn+MEPflDnMl27dr3l9WdkZPDaa6/hdruJjo6+7jLR0dHXvS0yMrJJfuiaar1er5clS5awZcsWANLT08nJySE5ObnRv1Zdmmq+QGL3Ge06X6T5xyHvyMgIW854hV234RV2nw/sP2NTzFff9VlafFJTU5v0TTC3b99OixYtblh67OLcuXO4XC5OnToFwNChQxk5ciTh4dqXLyIi8nVBc3Lz0aNHOX/+PEePHsXn87F9+3YAunXrRkJCAvPnz6ekpIR7772XmJgYioqKeP311/n5z39ubfAmtnPnThYsWEBNTQ1xcXHMmDGDbt26WR1LREQkIAVN8XnxxRf5+OOPay8PHDgQgJUrV3L//fcTGRnJ+++/z7PPPosxhm7duvHOO+/wxBNPWBW5SXk8HhYvXsznn38OQKdOncjOzm7U85JERETsJmiKz6xZs+p8DZ8JEyZc9cKFdnb27Fny8vI4ffo0AMOHD2fEiBF61paIiMhNBE3xkct27NjBwoUL8Xg8xMfHk52d/a1OABcREQklKj5BoqamhkWLFtWe29SlSxeys7NJSEiwNpiIiEgQUfEJAqdPn8blcnHmzBkcDgcjRozgvvvu06EtERGRBlLxCWDGGLZt28aiRYvwer0kJCSQk5ND586drY4mIiISlFR8ApTb7WbhwoXs3LkTgNtvv50ZM2YQHx9vcTIREZHgpeITgE6dOoXL5eLcuXM4HA5GjRrF0KFD63yzVREREbk5FZ8AYoxh69atLF68GJ/PR1JSEjk5OXTs2NHqaCIiIrag4hMg3G438+fPZ9euXQD06NGD6dOnB+07xIuIiAQiFZ8AcPLkSfLy8rhw4QJhYWGMHj2azMxMHdoSERFpZCo+FjLGsHnzZpYuXYrP5yM5ORmn00mHDh2sjiYiImJLKj4Wqa6uZt68eezZsweAXr16MW3aNGJjYy1OJiIiYl8qPhYoLi7G5XJRWlpKWFgY48aNY/DgwTq0JSIi0sRUfJqRMYZNmzaxYsUK/H4/LVq0wOl0kpaWZnU0ERGRkKDi00yqqqo4dOgQO3bsAKB3795MnTqVmJgYi5OJiIiEDhWfZnD+/Hk+/vhjysvLCQ8PZ/z48dx99906tCUiItLMVHyaQXJyMomJibjdbr73ve/pWVsiIiIW0dt7N4Pw8HCys7Pp0aMHbdu2tTqOiIhIyFLxaSaJiYmEh4dbHUNERCSkqfiIiIhIyFDxERERkZCh4iMiIiIhQ8VHREREQoaKj4iIiIQMFR8REREJGSo+IiIiEjJUfERERCRkqPiIiIhIyFDxERERkZCh4iMiIiIhQ8VHREREQoaKj4iIiISMCKsDBBpjDADl5eWNul6Px0NlZSXl5eVERkY26roDgd3nA/vPaPf5Kmu8+N2VwOXHt8PnsThR47P7NrT7fGD/GZtyvit/t6/8Hb8Rh7nZEiHm+PHjpKenWx1DREREbsGxY8fo0KHDDW9X8fkGv9/PiRMnSExMxOFwNNp6y8vLSU9P59ixYyQlJTXaegOF3ecD+89o9/nA/jNqvuBn9xmbcj5jDBUVFaSlpREWduMzeXSo6xvCwsLqbIrfVlJSki1/mK+w+3xg/xntPh/Yf0bNF/zsPmNTzZecnHzTZXRys4iIiIQMFR8REREJGSo+zSQ6OpqXXnqJ6Ohoq6M0CbvPB/af0e7zgf1n1HzBz+4zBsJ8OrlZREREQob2+IiIiEjIUPERERGRkKHiIyIiIiFDxUdERERChopPI/n1r3/NkCFDiIuLIyUl5brLHD16lMmTJxMXF0ebNm34xS9+gdfrrXO958+f57vf/S5JSUmkpKTw+OOPc/HixSaYoGE+/fRTHA7HdT82b958w/vdf//91yz/4x//uBmTN0znzp2vyfvmm2/WeZ/q6mqefvppWrVqRUJCAjk5OZSUlDRT4vo7fPgwjz/+OF26dCE2Npbbb7+dl156iZqamjrvF8jb8P3336dz587ExMSQkZHBpk2b6lw+Ly+PXr16ERMTQ79+/SgsLGympA33xhtvcM8995CYmEibNm3Iyspi3759dd5n1qxZ12yrmJiYZkrcMC+//PI1WXv16lXnfYJp+8H1f584HA6efvrp6y4f6Ntv9erVTJ06lbS0NBwOBwUFBVfdbozhxRdfpH379sTGxjJmzBj2799/0/U29HHcUCo+jaSmpobc3Fyeeuqp697u8/mYPHkyNTU1rFu3jo8//phZs2bx4osv1rne7373u+zatYuioiIWLFjA6tWrefLJJ5tihAYZMmQIJ0+evOrjhz/8IV26dOHuu++u875PPPHEVfd76623min1rXn11Vevyvs//sf/qHP5Z599lvnz55OXl8eqVas4ceIE2dnZzZS2/vbu3Yvf7+f3v/89u3bt4t133+WDDz7gf/7P/3nT+wbiNvzb3/7GzJkzeemll/j888/p378/48eP5/Tp09ddft26dTz44IM8/vjjbNu2jaysLLKysvjyyy+bOXn9rFq1iqeffpoNGzZQVFSEx+Nh3LhxXLp0qc77JSUlXbWtjhw50kyJG65Pnz5XZV2zZs0Nlw227QewefPmq+YrKioCIDc394b3CeTtd+nSJfr378/7779/3dvfeust/v3f/50PPviAjRs3Eh8fz/jx46murr7hOhv6OL4lRhrVRx99ZJKTk6+5vrCw0ISFhZlTp07VXve73/3OJCUlGbfbfd117d692wBm8+bNtdctWrTIOBwOU1xc3OjZv42amhqTmppqXn311TqXGzFihPnZz37WPKEaQadOncy7775b7+VLS0tNZGSkycvLq71uz549BjDr169vgoSN66233jJdunSpc5lA3YaDBw82Tz/9dO1ln89n0tLSzBtvvHHd5f/pn/7JTJ48+arrMjIyzI9+9KMmzdlYTp8+bQCzatWqGy5zo99Hgeill14y/fv3r/fywb79jDHmZz/7mbn99tuN3++/7u3BtP0AM2fOnNrLfr/ftGvXzvzbv/1b7XWlpaUmOjra/Pd///cN19PQx/Gt0B6fZrJ+/Xr69etH27Zta68bP3485eXl7Nq164b3SUlJuWoPypgxYwgLC2Pjxo1Nnrkh5s2bx7lz53j00Udvuuyf//xnWrduTd++fXnhhReorKxshoS37s0336RVq1YMHDiQf/u3f6vz8OTWrVvxeDyMGTOm9rpevXrRsWNH1q9f3xxxv5WysjJatmx50+UCbRvW1NSwdevWq77vYWFhjBkz5obf9/Xr11+1PFx+TAbDdoLL2wq46fa6ePEinTp1Ij09nenTp9/w900g2L9/P2lpaXTt2pXvfve7HD169IbLBvv2q6mp4U9/+hOPPfZYnW+IHUzb7+sOHTrEqVOnrtpGycnJZGRk3HAb3crj+FboTUqbyalTp64qPUDt5VOnTt3wPm3atLnquoiICFq2bHnD+1jlww8/ZPz48Td9g9eHHnqITp06kZaWxhdffMGvfvUr9u3bx+zZs5spacP89Kc/ZdCgQbRs2ZJ169bxwgsvcPLkSd55553rLn/q1CmioqKuOc+rbdu2AbfNvunAgQP89re/5e23365zuUDchmfPnsXn8133MbZ3797r3udGj8lA304Afr+fZ555hqFDh9K3b98bLtezZ0/++Mc/cuedd1JWVsbbb7/NkCFD2LVrV5O+GfOtyMjIYNasWfTs2ZOTJ0/yyiuvcN999/Hll1+SmJh4zfLBvP0ACgoKKC0t5Qc/+MENlwmm7fdNV7ZDQ7bRrTyOb4WKTx2ef/55fvOb39S5zJ49e256Al4wuZWZjx8/zpIlS/jkk09uuv6vn5/Ur18/2rdvz+jRozl48CC33377rQdvgIbMOHPmzNrr7rzzTqKiovjRj37EG2+8EbAvKX8r27C4uJgJEyaQm5vLE088Ued9A2Ebhrqnn36aL7/8ss5zYAAyMzPJzMysvTxkyBDuuOMOfv/73/Paa681dcwGmThxYu3nd955JxkZGXTq1IlPPvmExx9/3MJkTePDDz9k4sSJpKWl3XCZYNp+wUTFpw7PPfdcnW0coGvXrvVaV7t27a45M/3KM33atWt3w/t884Qur9fL+fPnb3ifb+tWZv7oo49o1aoV06ZNa/DXy8jIAC7vbWiuP5rfZrtmZGTg9Xo5fPgwPXv2vOb2du3aUVNTQ2lp6VV7fUpKSppsm31TQ+c7ceIEI0eOZMiQIfzhD39o8NezYht+U+vWrQkPD7/m2XN1fd/btWvXoOUDxU9+8pPaJzo09L/+yMhIBg4cyIEDB5ooXeNJSUmhR48eN8warNsP4MiRIyxbtqzBe0mDaftd2Q4lJSW0b9++9vqSkhIGDBhw3fvcyuP4ljTa2UJijLn5yc0lJSW11/3+9783SUlJprq6+rrrunJy85YtW2qvW7JkSUCd3Oz3+02XLl3Mc889d0v3X7NmjQHMjh07GjlZ0/jTn/5kwsLCzPnz5697+5WTm10uV+11e/fuDdiTm48fP266d+9uvvOd7xiv13tL6wiUbTh48GDzk5/8pPayz+czt912W50nN0+ZMuWq6zIzMwP25Fi/32+efvppk5aWZr766qtbWofX6zU9e/Y0zz77bCOna3wVFRWmRYsW5n//7/993duDbft93UsvvWTatWtnPB5Pg+4XyNuPG5zc/Pbbb9deV1ZWVq+TmxvyOL6lrI22phB35MgRs23bNvPKK6+YhIQEs23bNrNt2zZTUVFhjLn8A9u3b18zbtw4s337drN48WKTmppqXnjhhdp1bNy40fTs2dMcP3689roJEyaYgQMHmo0bN5o1a9aY7t27mwcffLDZ57uRZcuWGcDs2bPnmtuOHz9uevbsaTZu3GiMMebAgQPm1VdfNVu2bDGHDh0yc+fONV27djXDhw9v7tj1sm7dOvPuu++a7du3m4MHD5o//elPJjU11Xz/+9+vXeabMxpjzI9//GPTsWNHs2LFCrNlyxaTmZlpMjMzrRihTsePHzfdunUzo0ePNsePHzcnT56s/fj6MsGyDf/617+a6OhoM2vWLLN7927z5JNPmpSUlNpnUj788MPm+eefr11+7dq1JiIiwrz99ttmz5495qWXXjKRkZFm586dVo1Qp6eeesokJyebTz/99KptVVlZWbvMN2d85ZVXzJIlS8zBgwfN1q1bzXe+8x0TExNjdu3aZcUIdXruuefMp59+ag4dOmTWrl1rxowZY1q3bm1Onz5tjAn+7XeFz+czHTt2NL/61a+uuS3Ytl9FRUXt3zrAvPPOO2bbtm3myJEjxhhj3nzzTZOSkmLmzp1rvvjiCzN9+nTTpUsXU1VVVbuOUaNGmd/+9re1l2/2OG4MKj6N5JFHHjHANR8rV66sXebw4cNm4sSJJjY21rRu3do899xzVzX+lStXGsAcOnSo9rpz586ZBx980CQkJJikpCTz6KOP1papQPDggw+aIUOGXPe2Q4cOXfU9OHr0qBk+fLhp2bKliY6ONt26dTO/+MUvTFlZWTMmrr+tW7eajIwMk5ycbGJiYswdd9xhXn/99av20H1zRmOMqaqqMv/8z/9sWrRoYeLi4syMGTOuKhOB4qOPPrruz+zXdwQH2zb87W9/azp27GiioqLM4MGDzYYNG2pvGzFihHnkkUeuWv6TTz4xPXr0MFFRUaZPnz5m4cKFzZy4/m60rT766KPaZb454zPPPFP7/Wjbtq2ZNGmS+fzzz5s/fD088MADpn379iYqKsrcdttt5oEHHjAHDhyovT3Yt98VS5YsMYDZt2/fNbcF2/a78jfrmx9XZvD7/eZf//VfTdu2bU10dLQZPXr0NXN36tTJvPTSS1ddV9fjuDE4jDGm8Q6ciYiIiAQuvY6PiIiIhAwVHxEREQkZKj4iIiISMlR8REREJGSo+IiIiEjIUPERERGRkKHiIyIiIiFDxUdERERChoqPiIiIhAwVHxEREQkZKj4iIiISMlR8RMTWzpw5Q7t27Xj99ddrr1u3bh1RUVEsX77cwmQiYgW9SamI2F5hYSFZWVmsW7eOnj17MmDAAKZPn84777xjdTQRaWYqPiISEp5++mmWLVvG3Xffzc6dO9m8eTPR0dFWxxKRZqbiIyIhoaqqir59+3Ls2DG2bt1Kv379rI4kIhbQOT4iEhIOHjzIiRMn8Pv9HD582Oo4ImIR7fEREdurqalh8ODBDBgwgJ49e/Lee++xc+dO2rRpY3U0EWlmKj4iYnu/+MUvcLlc7Nixg4SEBEaMGEFycjILFiywOpqINDMd6hIRW/v000957733+K//+i+SkpIICwvjv/7rv/jss8/43e9+Z3U8EWlm2uMjIiIiIUN7fERERCRkqPiIiIhIyFDxERERkZCh4iMiIiIhQ8VHREREQoaKj4iIiIQMFR8REREJGSo+IiIiEjJUfERERCRkqPiIiIhIyFDxERERkZDx/wPixg68g3tt4QAAAABJRU5ErkJggg==", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(df.x, df.y, color=\"grey\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.grid()\n", "plt.axhline()\n", "plt.axvline()\n", "\n", "# set the slope\n", "m = 1.5\n", "\n", "# get the y-intercept\n", "yInt = -2\n", "\n", "# plot the slope from the y-intercept for 1x\n", "mx = [0, 1]\n", "my = [yInt, yInt + m]\n", "plt.plot(mx,my, color='red', lw=5)\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Slope-Intercept Form\n", "One of the great things about algebraic expressions is that you can write the same equation in multiple ways, or *forms*. The *slope-intercept form* is a specific way of writing a 2-variable linear equation so that the equation definition includes the slope and y-intercept. The generalised slope-intercept form looks like this:\n", "\n", "\\begin{equation}y = mx + b \\end{equation}\n", "\n", "In this notation, ***m*** is the slope and ***b*** is the y-intercept.\n", "\n", "For example, let's look at the solved linear equation we've been working with so far in this section:\n", "\n", "\\begin{equation}y = \\frac{3x - 4}{2} \\end{equation}\n", "\n", "Now that we know the slope and y-intercept for the line that this equation defines, we can rewrite the equation as:\n", "\n", "\\begin{equation}y = 1\\frac{1}{2}x + -2 \\end{equation}\n", "\n", "You can see intuitively that this is true. In our original form of the equation, to find y we multiply x by three, subtract 4, and divide by two - in other words, x is half of 3x - 4; which is 1.5x - 2. So these equations are equivalent, but the slope-intercept form has the advantages of being simpler, and including two key pieces of information we need to plot the line represented by the equation. We know the y-intecept that the line passes through (0, -2), and we know the slope of the line (for every x, we add 1.5 to y.\n", "\n", "Let's recreate our set of test x and y values using the slope-intercept form of the equation, and plot them to prove that this describes the same line:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "\n", "import pandas as pd\n", "from matplotlib import pyplot as plt\n", "\n", "# Create a dataframe with an x column containing values from -10 to 10\n", "df = pd.DataFrame ({'x': range(-10, 11)})\n", "\n", "# Define slope and y-intercept\n", "m = 1.5\n", "yInt = -2\n", "\n", "# Add a y column by applying the slope-intercept equation to x\n", "df['y'] = m*df['x'] + yInt\n", "\n", "# Plot the line\n", "from matplotlib import pyplot as plt\n", "\n", "plt.plot(df.x, df.y, color=\"grey\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.grid()\n", "plt.axhline()\n", "plt.axvline()\n", "\n", "# label the y-intercept\n", "plt.annotate('y-intercept',(0,yInt))\n", "\n", "# plot the slope from the y-intercept for 1x\n", "mx = [0, 1]\n", "my = [yInt, yInt + m]\n", "plt.plot(mx,my, color='red', lw=5)\n", "\n", "plt.show()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.9" } }, "nbformat": 4, "nbformat_minor": 2 }