<\r\n" );
document.write( "I need help with these problems, most of them do need to be graphed. I want to\r\n" );
document.write( "thank the person who helps out. Thank you and I really am grateful. \r\n" );
document.write( "Solve the system by graphing.\r\n" );
document.write( "\r\n" );
document.write( "3x+y=6\r\n" );
document.write( "3x-y=0 \r\n" );
document.write( "\r\n" );
document.write( "The intercepts of the first line are (0,6) and (2,0). So plot\r\n" );
document.write( "those two points and draw a line through them:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "The only intercept of the second line is (0,0) so we must find \r\n" );
document.write( "another point. Choose, say x = 1 then\r\n" );
document.write( " 3x - y = 0\r\n" );
document.write( "3(1) - y = 0 \r\n" );
document.write( " 3 - y = 0\r\n" );
document.write( " -y = -3\r\n" );
document.write( " y = 3\r\n" );
document.write( "\r\n" );
document.write( "So plot (0,0) and (1,3) and draw a green line through them\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Through the point where those two line cross, draw a\r\n" );
document.write( "dotted horizontal line\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Through the point where those two line cross, draw a\r\n" );
document.write( "horizontal line\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Also draw a vertical line through the point where they cross:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Notice that the vertical line cuts the x-axis at 1, and\r\n" );
document.write( "the horizontal line cuts the y-axis at 3. So the solution of the system\r\n" );
document.write( "is (x,y) = (1,3)\r\n" );
document.write( "\r\n" );
document.write( "Check by substituting into both equations:\r\n" );
document.write( "\r\n" );
document.write( "Substituting in the first equation:\r\n" );
document.write( "\r\n" );
document.write( " 3x + y = 6\r\n" );
document.write( "3(1) + 3(3) = 6\r\n" );
document.write( " 3 + 3 = 6\r\n" );
document.write( " 6 = 6\r\n" );
document.write( "\r\n" );
document.write( "That's true. \r\n" );
document.write( "\r\n" );
document.write( "Substituting in the first equation:\r\n" );
document.write( "\r\n" );
document.write( " 3x - y = 0\r\n" );
document.write( " 3(1) - (3) = 0\r\n" );
document.write( " 3 - 3 = 0\r\n" );
document.write( " 0 = 0\r\n" );
document.write( "\r\n" );
document.write( "That's also true. So we know (x,y) = (1,3)\r\n" );
document.write( "is the correct solution to the system\r\n" );
document.write( "\r\n" );
document.write( "3x + y = 6\r\n" );
document.write( "3x - y = 0 \r\n" );
document.write( "\r\n" );
document.write( "--------------------------\r\n" );
document.write( "\r\n" );
document.write( "Solve the following system of linear inequalities by graphing.\r\n" );
document.write( "\r\n" );
document.write( "3x - y < 2\r\n" );
document.write( " x + y < 2 \r\n" );
document.write( "\r\n" );
document.write( "We draw the boundary line for the first one. We draw it\r\n" );
document.write( "dotted because it's \"<\" and not \"<\". The equation\r\n" );
document.write( "of the boundary line is 3x - y = 2.\r\n" );
document.write( "\r\n" );
document.write( "Two points on it are (0,-2) and (1,1). So we plot those\r\n" );
document.write( "and draw a dotted line through them:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "No we choose a test point which can be any point not on the line.\r\n" );
document.write( "We pick the point (0,0) since it is the easiest point not on the line.\r\n" );
document.write( "\r\n" );
document.write( "We substitute it in the INEQUALITY\r\n" );
document.write( "\r\n" );
document.write( " 3x - y < 2\r\n" );
document.write( "3(0) - (0) < 2\r\n" );
document.write( " 0 < 2\r\n" );
document.write( "\r\n" );
document.write( "This is true, so we will shade the side of the line which the\r\n" );
document.write( "origin (0,0) is on. That means we will shade the LEFT (upper) side of\r\n" );
document.write( "the line because the test point, which is the origin (0,0), is\r\n" );
document.write( "a point on the LEFT (upper) side of the line which contains the solutions\r\n" );
document.write( "to the inequality. \r\n" );
document.write( "\r\n" );
document.write( "Now we draw the boundary line for the second one. We also draw it\r\n" );
document.write( "dotted because it's \"<\" and not \"<\". The equation\r\n" );
document.write( "of the boundary line is x + y = 2.\r\n" );
document.write( "\r\n" );
document.write( "Two points on it are (0,2) and (2,0). So we plot those\r\n" );
document.write( "and draw a dotted line through them:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "No we choose a test point which can be any point not on the line.\r\n" );
document.write( "We pick the point (0,0) again since it is the easiest point not on \r\n" );
document.write( "the line.\r\n" );
document.write( "\r\n" );
document.write( "We substitute it in the INEQUALITY\r\n" );
document.write( "\r\n" );
document.write( " x + y < 2\r\n" );
document.write( " 0 + 0 < 2\r\n" );
document.write( " 0 < 2\r\n" );
document.write( "\r\n" );
document.write( "This is true, so we will shade the side of the line which the\r\n" );
document.write( "origin (0,0) is on. That means we will shade the lower or LEFT side of\r\n" );
document.write( "the line because the test point, which is the origin (0,0), is\r\n" );
document.write( "a point on the lower LEFT side of the line which contains the solutions\r\n" );
document.write( "to the inequality. \r\n" );
document.write( "\r\n" );
document.write( "Now to shade the solution set. We shade the region which is LEFT of\r\n" );
document.write( "both lines. \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Solve the following system of linear inequalities bt graphing\r\n" );
document.write( "x - y < 3\r\n" );
document.write( "x + 2y > 6\r\n" );
document.write( "\r\n" );
document.write( "Draw the first boundary line dotted:\r\n" );
document.write( "\r\n" );
document.write( "x - y = 3 \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "The origin (0,0) as a test point satisfies x - y < 3, and so\r\n" );
document.write( "we will shade the LEFT (upper) side of that line, since the origin is\r\n" );
document.write( "on that side.\r\n" );
document.write( "\r\n" );
document.write( "Draw the second boundary line dotted:\r\n" );
document.write( "\r\n" );
document.write( "x + 2y = 6 \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "The origin (0,0) as a test point DOES NOT satisfy x + 2y > 6, and so\r\n" );
document.write( "we will shade the RIGHT (upper) side of that green line, since the \r\n" );
document.write( "origin is NOT on that side of the green line.\r\n" );
document.write( "\r\n" );
document.write( "So we shad the region which is right of the green line and left of the\r\n" );
document.write( "dark red line:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "(x-3)(sqrt(sin(7x)))/( sqrt( sin(7x) ) ),((6-x)/2)(sqrt(sin(7x)))/( sqrt( sin(7x) ) ) )}}}\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Solve the system by addition.\r\n" );
document.write( "\r\n" );
document.write( "5x - 4y = -5\r\n" );
document.write( " x - 2y = -7 \r\n" );
document.write( "\r\n" );
document.write( "To make the y's cancel, multiply the bottom equation by -2,\r\n" );
document.write( "then add vertically term by term:\r\n" );
document.write( "\r\n" );
document.write( " 5x - 4y = -5\r\n" );
document.write( "-2x + 4y = 14\r\n" );
document.write( "-------------\r\n" );
document.write( " 3x = 9\r\n" );
document.write( " x = 3\r\n" );
document.write( "\r\n" );
document.write( "To make the x's cancel, multiply the bottom equation by -5,\r\n" );
document.write( "then add vertically term by term:\r\n" );
document.write( "\r\n" );
document.write( " 5x - 4y = -5\r\n" );
document.write( "-5x + 10y = 35\r\n" );
document.write( "--------------\r\n" );
document.write( " 6y = 30\r\n" );
document.write( " y = 5\r\n" );
document.write( "\r\n" );
document.write( "So the solution is (x,y) = (3,5)\r\n" );
document.write( "\r\n" );
document.write( "That one could have been solved graphically:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Then draw a horizontal line and a vertical line through the \r\n" );
document.write( "point where the lines cross:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "and you see that the vertical line crosses the x-axis at 3,\r\n" );
document.write( "and the horizontal line crosses the y-axis at 5. So we get\r\n" );
document.write( "the same solution (x,y) = (3,5) by graphing that we get by\r\n" );
document.write( "addition.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "
\n" );
document.write( " \n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
