document.write( "Question 810282: Please help. I need to graph this inequalities and fine the feasible region. The inequalities are; −3≤y≤5
\n" ); document.write( "4x+y≤5
\n" ); document.write( "−2x+y≤5.
\n" ); document.write( "and says E = 4x - 3y.
\n" ); document.write( "My teacher gets mad it homework is wrong but mainly if not done. Thanks. \n" ); document.write( "
Algebra.Com's Answer #488082 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
I will show you how to graph those inequalities and find the feasible region first.
\n" ); document.write( "After that, I will tell you what I think of \"E+=+4x+-+3y\" .
\n" ); document.write( "
\n" ); document.write( "THE FEASIBLE REGION:
\n" ); document.write( "The borders of your feasible region may be part of the lines
\n" ); document.write( "\"y=-3\" , \"y=5\" ,
\n" ); document.write( "\"4x%2By=5\" , and
\n" ); document.write( "\"-2x%2By=5\"
\n" ); document.write( "Each of those lines is part of the solution to one of the inequalities.
\n" ); document.write( "The lines represented by \"y=-3\" and \"y=5\" are easy to figure out.
\n" ); document.write( "They are \"horizontal\" lines, where all the points have the same y-coordinate:
\n" ); document.write( "\"graph%28300%2C210%2C-10%2C10%2C-6%2C8%2C9%2C-3%2C5%29\"
\n" ); document.write( "The other two lines are slanted lines that are a little harder to figure out.
\n" ); document.write( "However, since two points determine a line, all we need is two find 2 points for each one.
\n" ); document.write( "Also, since they have to cross the horizontal lines \"y=-3\" and \"y=5\" ,
\n" ); document.write( "we might as well find the point where they cross \"y=-3\" and \"y=5\" .
\n" ); document.write( "To find each intersection point we solve a system of equations, but these are very simple.
\n" ); document.write( "Where does the horizontal line \"y=5\" cross the slanted lines?
\n" ); document.write( "\"system%28y=5%2C4x%2By=5%29\" --> \"4x%2B5=5\" --> \"4x=0\" --> \"x=0\" gives us point {0,5) .
\n" ); document.write( "\"system%28y=5%2C-2x%2By=5%29\" --> \"-2x%2B5=5\" --> \"-2x=0\" --> \"x=0\" gives us point {0,5) too.
\n" ); document.write( "So, it turns out that the 3 lines represented by
\n" ); document.write( "\"y=5\" , \"4x%2By=5\" , and \"-2x%2By=5\" all cross at point \"A%280%2C5%29\" .
\n" ); document.write( "Where does the horizontal line \"y=-3\" cross the slanted lines?
\n" ); document.write( "\"system%28y=-3%2C4x%2By=5%29\" --> \"4x-3=5\" --> \"4x=3%2B5\" --> \"4x=8\" --> \"x=8%2F4\" --> \"x=2\" gives us point \"B%282%2C-3%29\" .
\n" ); document.write( "\"system%28y=-3%2C-2x%2By=5%29\" --> \"-2x-3=5\" --> \"-2x=3%2B5\" --> \"-2x=8\" --> \"x=8%2F%28-2%29\" --> \"x=-4\" gives us point \"C%28-4%2C-3%29\" .
\n" ); document.write( "Now we can mark those points and draw the slanted lines:
\n" ); document.write( "\"4x%2By=5\" , passing through (0,5) and {2,-3) , and
\n" ); document.write( "\"-2x%2By=5\" , passing through (0,5) and {-4,-3) .
\n" ); document.write( " Is triangle ABC the feasible region?
\n" ); document.write( "(Feasible regions are often polygons, like triangles, quadrilaterals, maybe even pentagons).
\n" ); document.write( "If triangle ABC is the feasible region, the origin, point \"O%280%2C0%29\" should be a solution of all the inequalities.
\n" ); document.write( "\"-3%3C0%3C5\" , \"4%2A0%2B0=0%3C5\" , and (((-2*0+0=0<5}}} ,
\n" ); document.write( "so point \"O%280%2C0%29\" is a solution of all the inequalities.
\n" ); document.write( "For each of the 4 inequalities that determine the feasible region,
\n" ); document.write( "\"-3%3C=y\", \"y%3C=5\", \"4x%2By=5\" , and \"-2x%2By=5\" ,
\n" ); document.write( "the solution is the corresponding boundary line, plus the whole side of the line that contains the origin, point \"O%280%2C0%29\" .
\n" ); document.write( "So the points that satisfy all 4 inequalities are the points in triangle ABC, including the sides of the triangle.
\n" ); document.write( "
\n" ); document.write( "WHAT ABOUT \"E+=+4x+-+3y\" ?
\n" ); document.write( "Usually you are asked to find out where in the feasible region the function \"E\" has a maximum or minimum value.
\n" ); document.write( "(E is a linear function of x and y).
\n" ); document.write( "For a linear function in a polygon-shaped feasible region,
\n" ); document.write( "the maximum must happen in one of the vertices or all along one tof the sides/edges of the feasible region.
\n" ); document.write( "The sames goes for the minimum.
\n" ); document.write( "All we have to do is calculate the value of \"E\" at points A, B, and C
\n" ); document.write( "At \"A%280%2C5%29\" ,
\n" ); document.write( "\"E=4%2A0-3%2A5=-15\"
\n" ); document.write( "At \"B%282%2C-3%29\" ,
\n" ); document.write( "\"E=4%2A2-3%2A%28-3%29=8%2B9=17\"
\n" ); document.write( "At \"C%28-4%2C-3%29\" ,
\n" ); document.write( "\"E=4%2A%28-4%29-3%2A%28-3%29=-16%2B9=-7\"
\n" ); document.write( "So the maximum of E happens at \"B%282%2C-3%29\" , where \"E=17\" ,
\n" ); document.write( "and the minimum happens at \"A%280%2C5%29\" , where \"E=-15\" . \n" ); document.write( "
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