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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "Use graphing to solve this system of equations:\r\n" );
document.write( "y = 3x + 2 \r\n" );
document.write( "y = -x - 2 \r\n" );
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document.write( "A SOLUTION:\r\n" );
document.write( "When you use a graphing approach, you are going to make a graph with an x- and y-axis and plot the \r\n" );
document.write( "graphs of these equations. \r\n" );
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document.write( "These happen to be two linear equations so their graphs will be lines. Except for two special cases (that \r\n" );
document.write( "don't apply here), the solution to a system of linear equations is the point where the two lines intersect. \r\n" );
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document.write( "Your equations are both in slope-intercept form (y=mx+b). This makes them pretty easy to graph.\r\n" );
document.write( "In slope-intercept form, the number in front of x (the x-coefficient) is the slope of the line. The constant \r\n" );
document.write( "term to the far right, is the y-intercept. \r\n" );
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document.write( "The y-intercept is the place where the line crosses the y-axis (the vertical axis).\r\n" );
document.write( "The slope tells you how steep the line is, the change in y divided by the change in x.\r\n" );
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document.write( "Look at the first equation, y = 3x + 2,\r\n" );
document.write( "The slope is 3;\r\n" );
document.write( "The y-intercept is 2.\r\n" );
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document.write( "For the second equation, y = -x - 2,\r\n" );
document.write( "The slope is -1;\r\n" );
document.write( "The y-intercept is -2.\r\n" );
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document.write( "Now we are ready to graph. Using graph paper, draw your x- and y-axes. Like this:\r\n" );
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document.write( "To graph the first equation, y = 3x + 2, find the point (0,2). This is the y-intercept. The line will cross the y-axis here. Mark this point.\r\n" );
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document.write( "The slope is 3. This means the line goes up 3 units for each step to the right.\r\n" );
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document.write( "Start at (0,2). Count up 3 units. Then count 1 unit to the right. Make a mark there at (1,5). \r\n" );
document.write( "Repeat this one more time, starting at your secant point (up 3, right 1). Mark the point.\r\n" );
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document.write( "All three points you marks=ed are on the line. Use your ruler to carefully draw.\r\n" );
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document.write( "It should look like this:\r\n" );
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\r\n" );
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document.write( "Now we are ready to graph the second equation, y=-x - 2:\r\n" );
document.write( "Mark at the y-intercept at (0,-2)\r\n" );
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document.write( "This time we have a slope of -1. When the slope is negative, the line goes down for each step to the right.\r\n" );
document.write( "Starting at the y-intercept, count down 1, right one and mark the point (1, -3).\r\n" );
document.write( "Repeat this several times because the points are so close together.\r\n" );
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document.write( "Your graph should look like this:\r\n" );
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\r\n" );
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document.write( "Solving by graphing is an estimating process. It looks like the interception point is very close to (-1, -1). This is the solution by graphing.\r\n" );
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document.write( "CHECK:\r\n" );
document.write( "If you substitute x= -1 and y= -1 into both equations, you will see that this is a very good estimate.\r\n" );
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document.write( "y = 3x + 2 \r\n" );
document.write( "(-1) = 3(-1) + 2\r\n" );
document.write( "-1 = -3 + 2\r\n" );
document.write( "-1 = -1\r\n" );
document.write( "TRUE\r\n" );
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document.write( "y = -x - 2 \r\n" );
document.write( "(-1) = -(-1) - 2\r\n" );
document.write( "-1 = 1 - 2\r\n" );
document.write( "-1 = -1\r\n" );
document.write( "TRUE\r\n" );
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document.write( "The ordered pair (-1, -1) is exactly the solution to this system of equations.\r\n" );
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document.write( "Hope this helps! Feel free to email if you have ANY questions about the solution.\r\n" );
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document.write( "Good luck with your math,\r\n" );
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document.write( "Mrs. F\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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