document.write( "Question 49153: solve & graph the equations\r
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\n" ); document.write( "\n" ); document.write( "equation 1 x-axis intercept=
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Algebra.Com's Answer #32567 by AnlytcPhil(1806) $\"\"$ $\"About$
You can put this solution on YOUR website!
solve & graph the \r\n" );
document.write( "equations \r\n" );
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document.write( "3x - 6y =  12\r\n" );
document.write( "2x - 5y = -10 \r\n" );
document.write( "\r\n" );
document.write( "To eliminate the y's:\r\n" );
document.write( "The coefficients of y in absolute\r\n" );
document.write( "value are 6 and 5.  The smallest\r\n" );
document.write( "positive integer which both 6 and 5\r\n" );
document.write( "will divide evenly into is 30. So\r\n" );
document.write( "we need to multiply each equation\r\n" );
document.write( "thru by whatever will cause one\r\n" );
document.write( "coefficient of y to be 30, and the\r\n" );
document.write( "other coefficient of y to be -30.\r\n" );
document.write( "There are two choices.  We can\r\n" );
document.write( "either multiply the first equation\r\n" );
document.write( "thru by -5 and the second thru by\r\n" );
document.write( "+6 or the first thru by +5 and the\r\n" );
document.write( "second by -6.  I'll arbitrarily\r\n" );
document.write( "choose to multiply the first one\r\n" );
document.write( "thru by +5 and the second thru by\r\n" );
document.write( "-6:\r\n" );
document.write( "\r\n" );
document.write( " 5[3x - 6y =  12]\r\n" );
document.write( "-6[2x - 5y = -10]\r\n" );
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document.write( "Multiply thru, then add vertically\r\n" );
document.write( "term by term:\r\n" );
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document.write( "  15x - 30y =  60\r\n" );
document.write( " -12x + 30y =  60\r\n" );
document.write( " ----------------- \r\n" );
document.write( "   3x       = 120\r\n" );
document.write( "          x = 40\r\n" );
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document.write( "------------------------------\r\n" );
document.write( "\r\n" );
document.write( "3x - 6y =  12\r\n" );
document.write( "2x - 5y = -10 \r\n" );
document.write( "\r\n" );
document.write( "To eliminate the x's:\r\n" );
document.write( "The coefficients of x in absolute\r\n" );
document.write( "value are 3 and 2.  The smallest\r\n" );
document.write( "positive integer which both 3 and 2\r\n" );
document.write( "will divide evenly into is 6. So\r\n" );
document.write( "we need to multiply each equation\r\n" );
document.write( "thru by whatever will cause one\r\n" );
document.write( "coefficient of x to be 6, and the\r\n" );
document.write( "other coefficient of x to be -6.\r\n" );
document.write( "There are two choices.  We can\r\n" );
document.write( "either multiply the first equation\r\n" );
document.write( "thru by -2 and the second thru by\r\n" );
document.write( "+3 or the first thru by +2 and the\r\n" );
document.write( "second by -3.  I'll arbitrarily\r\n" );
document.write( "choose to multiply the first one\r\n" );
document.write( "thru by -2 and the second thru by\r\n" );
document.write( "+3:\r\n" );
document.write( "\r\n" );
document.write( "-2[3x - 6y =  12]\r\n" );
document.write( " 3[2x - 5y = -10]\r\n" );
document.write( "\r\n" );
document.write( "Multiply thru, then add vertically\r\n" );
document.write( "term by term:\r\n" );
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document.write( "  -6x + 12y = -24\r\n" );
document.write( "   6x - 15y = -30\r\n" );
document.write( " ----------------- \r\n" );
document.write( "        -3y = -54\r\n" );
document.write( "          y = 18\r\n" );
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document.write( "----------------------------\r\n" );
document.write( "\r\n" );
document.write( "x = 40\r\n" );
document.write( "y = 18 \r\n" );
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document.write( "So the two lines will\r\n" );
document.write( "cross at the point (40,18)\r\n" );
document.write( "---------------------------\r\n" );
document.write( "\r\n" );
document.write( "To find the x-coordinate\r\n" );
document.write( "of the x-axis intercept, \r\n" );
document.write( "substitute 0 for y and\r\n" );
document.write( "solve for x.\r\n" );
document.write( "\r\n" );
document.write( "To find the y-coordinate\r\n" );
document.write( "of the y-axis intercept, \r\n" );
document.write( "substitute 0 for x and\r\n" );
document.write( "solve for y.\r\n" );
document.write( "\r\n" );
document.write( "For equation 1,\r\n" );
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document.write( "3x - 6y =  12\r\n" );
document.write( "\r\n" );
document.write( "substitute 0 for y\r\n" );
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document.write( "3x - 6(0) = 12\r\n" );
document.write( "   3x - 0 = 12\r\n" );
document.write( "       3x = 12\r\n" );
document.write( "        x = 4\r\n" );
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document.write( "So the x-coordinate of\r\n" );
document.write( "the x-axis intercept is 4.\r\n" );
document.write( "The x-axis intercept is \r\n" );
document.write( "the point (4,0).  That's\r\n" );
document.write( "the point at which the\r\n" );
document.write( "graph of equation 1\r\n" );
document.write( "crosses the x-axis\r\n" );
document.write( "\r\n" );
document.write( "equation 1 x-axis intercept = (4,0)\r\n" );
document.write( "\r\n" );
document.write( "For equation 1,\r\n" );
document.write( "\r\n" );
document.write( "3x - 6y =  12\r\n" );
document.write( "\r\n" );
document.write( "substitute 0 for x\r\n" );
document.write( "\r\n" );
document.write( "3(0) - 6y = 12\r\n" );
document.write( "  0  - 6y = 12\r\n" );
document.write( "      -6y = 12\r\n" );
document.write( "        y = -2\r\n" );
document.write( "\r\n" );
document.write( "So the y-coordinate of\r\n" );
document.write( "the y-axis intercept is -2.\r\n" );
document.write( "The y-axis intercept is \r\n" );
document.write( "the point (0,-2).  That's\r\n" );
document.write( "the point at which the\r\n" );
document.write( "graph of equation 1\r\n" );
document.write( "crosses the y-axis.\r\n" );
document.write( "\r\n" );
document.write( "equation 1 y-axis intercept = (0,-2) \r\n" );
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document.write( "For equation 2,\r\n" );
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document.write( "2x - 5y = -10\r\n" );
document.write( "\r\n" );
document.write( "substitute 0 for y\r\n" );
document.write( "\r\n" );
document.write( "2x - 5(0) = -10\r\n" );
document.write( "   2x - 0 = -10\r\n" );
document.write( "       2x =  10\r\n" );
document.write( "        x = 5\r\n" );
document.write( "\r\n" );
document.write( "So the x-coordinate of\r\n" );
document.write( "the x-axis intercept is 5.\r\n" );
document.write( "The x-axis intercept is \r\n" );
document.write( "the point (5,0).  That's\r\n" );
document.write( "the point at which the\r\n" );
document.write( "graph of equation 2\r\n" );
document.write( "crosses the x-axis.\r\n" );
document.write( "\r\n" );
document.write( "equation 2 x-axis intercept = (5,0)\r\n" );
document.write( "\r\n" );
document.write( "For equation 2,\r\n" );
document.write( "\r\n" );
document.write( "2x - 5y = -10\r\n" );
document.write( "\r\n" );
document.write( "substitute 0 for x\r\n" );
document.write( "\r\n" );
document.write( "2(0) - 5y = -10\r\n" );
document.write( "  0  - 5y = -10\r\n" );
document.write( "      -5y = -10\r\n" );
document.write( "        y = 2\r\n" );
document.write( "\r\n" );
document.write( "So the y-coordinate of\r\n" );
document.write( "the y-axis intercept is 2.\r\n" );
document.write( "The y-axis intercept is \r\n" );
document.write( "the point (0,2).  That's\r\n" );
document.write( "the point at which the\r\n" );
document.write( "graph of equation 2\r\n" );
document.write( "crosses the y-axis.\r\n" );
document.write( "\r\n" );
document.write( "equation 2 y-axis intercept = (0,2) \r\n" );
document.write( "\r\n" );
document.write( "To graph the equations, get some\r\n" );
document.write( "points.  You already have two on\r\n" );
document.write( "each line.  Trouble is, the point\r\n" );
document.write( "where the lines cross has such\r\n" );
document.write( "large coefficients (40,18).  The\r\n" );
document.write( "graph will have to be so big that\r\n" );
document.write( "we will need some other points\r\n" );
document.write( "since those are so small relative\r\n" );
document.write( "to the size we'll have to make the\r\n" );
document.write( "graph.  So let's get some other\r\n" );
document.write( "points with bigger numbers:\r\n" );
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document.write( "equation 1     \r\n" );
document.write( "x  |  y        \r\n" );
document.write( "-------        \r\n" );
document.write( "24 | 10        \r\n" );
document.write( "30 | 13       \r\n" );
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document.write( "So we plot (24,10) and\r\n" );
document.write( "(30,13)\r\n" );
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document.write( "and we get this line\r\n" );
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document.write( "Then for equation 2, we find a\r\n" );
document.write( "couple of points with large\r\n" );
document.write( "coefficients:\r\n" );
document.write( "\r\n" );
document.write( "equation 2\r\n" );
document.write( "x  |  y\r\n" );
document.write( "-------\r\n" );
document.write( "20 | 10\r\n" );
document.write( "35 | 16\r\n" );
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document.write( "So we plot (20,10) and\r\n" );
document.write( "(35,16)\r\n" );
document.write( "\r\n" );
document.write( "and we draw that line on the\r\n" );
document.write( "same set of axes\r\n" );
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document.write( "and we see that they cross at the\r\n" );
document.write( "point (x,y) = (40,18).\r\n" );
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document.write( "Edwin
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