document.write( "Question 670856: In triangle DEF, the equation of the altitude from D through EF is 4x + 3y - 13 = 0 and the equation of the altitude from E through DF is 2x - y - 4 = 0. What do you call the point where these lines intersect? Find the point where these lines intersect. Explain each step of your solution. \n" ); document.write( "
Algebra.Com's Answer #417155 by Theo(13342)\"\" \"About 
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basically you're talking about the intersection of 2 lines whose equations are:
\n" ); document.write( "4x + 3y - 13 = 0
\n" ); document.write( "2x - y - 4 = 0
\n" ); document.write( "the fact that these lines are altitudes to sides of a triangle doesn't change the fact that you are simply looking for the point of their intersection.
\n" ); document.write( "if there was additional information they were looking for, it would make a difference, but the way the problem is structured, all that other information is extraneous to the heart of the problem which is to find the intersection of the 2 lines.
\n" ); document.write( "the 2 equations are, once again:
\n" ); document.write( "4x + 3y - 13 = 0
\n" ); document.write( "2x - y - 4 = 0
\n" ); document.write( "you need to solve these equations simultaneously to get the solution.
\n" ); document.write( "you need to first transform them to standard form which is ax + by = c
\n" ); document.write( "4x + 3y - 13 = 0 becomes 4x + 3y = 13 once you convert it.
\n" ); document.write( "2x - y - 4 = 0 becomes 2x - y = 4 once you convert it.
\n" ); document.write( "your 2 equations are now:
\n" ); document.write( "4x + 3y = 13
\n" ); document.write( "2x - y = 4
\n" ); document.write( "we will solve by elimination.
\n" ); document.write( "multiply the second equation by 2 to get:
\n" ); document.write( "4x - 2y = 8
\n" ); document.write( "your 2 equations are now:
\n" ); document.write( "4x + 3y = 13
\n" ); document.write( "4x - 2y = 8
\n" ); document.write( "subtract the second equation from the first equation to get:
\n" ); document.write( "5y = 5
\n" ); document.write( "divide both sides of this equation by 5 to get:
\n" ); document.write( "y = 1
\n" ); document.write( "substitute for y in either original equation to find the value of x.
\n" ); document.write( "we'll use:
\n" ); document.write( "4x + 3y = 13
\n" ); document.write( "replace y with 1 to get:
\n" ); document.write( "4x + 3 = 13
\n" ); document.write( "subtract 3 from both sides of this equation to get:
\n" ); document.write( "4x = 10
\n" ); document.write( "divide both sides of this equation by 4 to get:
\n" ); document.write( "x = 2.5
\n" ); document.write( "your solution should be:
\n" ); document.write( "x = 2.5
\n" ); document.write( "y = 1
\n" ); document.write( "this solution should apply to both equation.
\n" ); document.write( "the first original equation is:
\n" ); document.write( "4x + 3y = 13
\n" ); document.write( "replace x with 2.5 and y with 1 to gtet:
\n" ); document.write( "4(2.5) + 3(1) = 13 which becomes:
\n" ); document.write( "10 + 3 = 13 which is true.
\n" ); document.write( "the second original equation is:
\n" ); document.write( "2x - y = 4
\n" ); document.write( "replace x with 2.5 and y with 1 to get:
\n" ); document.write( "2(2.5) - 1) = 4 which becomes:
\n" ); document.write( "5 - 1 = 4 which is also true.
\n" ); document.write( "the solution applies to both equations and is confirmed as good.
\n" ); document.write( "the point where these lines intersect is (x,y) = (2.5,1).
\n" ); document.write( "that intersection point is the solution that is common to both equations.
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