document.write( "Question 158640This question is from textbook Algebra and Trigonometry Structure and Method book 2
\n" ); document.write( ": I have been working on this problem and I can't figure it out. I was wondering if someone could help me? Please and Thank you!! I deeply appreciate it!!\r
\n" ); document.write( "\n" ); document.write( "Find and equation in standard form of the line containing the given points.
\n" ); document.write( "(3/4,5/4),(-1/4,1/2) \n" ); document.write( "
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i would start by finding the equation in slope-intercept form.
\n" ); document.write( "general form of that equation is y = m*x + b
\n" ); document.write( "where m is the slope and b is the y-intercept when x = 0.
\n" ); document.write( "the slope is determined by (y2-y1)/(x2-x1)
\n" ); document.write( "y2 = 1/2
\n" ); document.write( "y1 = 5/4
\n" ); document.write( "x2 = -1/4
\n" ); document.write( "x1 = 3/4
\n" ); document.write( "y2-y1 = 1/2 - 5/4 = 2/4 - 5/4 = -3/4
\n" ); document.write( "x2-x1 = -1/4 - 3/4 = -4/4 = -1
\n" ); document.write( "slope becomes (-3/4) / -1 becomes 3/4
\n" ); document.write( "so part of the equation is y = (3/4)*x + b
\n" ); document.write( "to find b, we would use the formula to solve for one of the known coordinates.
\n" ); document.write( "using one of the given points of (3/4,5/4),
\n" ); document.write( "5/4 = 3/4*x + b becomes 5/4 = 3/4*(3/4) + b becomes 5/4 = 9/16 + b.
\n" ); document.write( "solving for b, the equation becomes b = 5/4 - 9/16 = 20/16 - 9/16 = 11/16.
\n" ); document.write( "looks like the b value is 11/16.
\n" ); document.write( "the slope intercept form of the equation then becomes y = 3/4*x + 11/16
\n" ); document.write( "to prove this is correct, take the same point and calculate the value of y given the value of x = 3/4.
\n" ); document.write( "we get y = (3/4)*(3/4) + 11/16 = 9/16 + 11/16 = 20/16.
\n" ); document.write( "the given value of y at that point is 5/4 which is the same as 20/16.
\n" ); document.write( "looks like the b value in the slope intercept formula is now correct.
\n" ); document.write( "calculating the value of y in the second set of coordinates given, we get
\n" ); document.write( "y = (3/4)*(-1/4) + 11/16
\n" ); document.write( "this becomes y = -3/16 + 11/16 = 8/16
\n" ); document.write( "the given y coordinate of 1/2 is the same as 8/16 so the formula looks good again.
\n" ); document.write( "now that we have the complete slope intercept form of the equation, we can translate to get the standard form of the equation.
\n" ); document.write( "the standard form of the equation is a*x + b*y = c
\n" ); document.write( "we start with the slope intercept form of the equation y = (3/4)*x + 11/16
\n" ); document.write( "we multiply both sides of the equation by 16 to remove the fractional parts.
\n" ); document.write( "the equation becomes 16*y = 12*x + 11
\n" ); document.write( "we subtract (12*x) from both sides of the equation to get 16*y - 12*x = 11
\n" ); document.write( "the standard form of the equation becomes -12*x + 16*y = 11
\n" ); document.write( "to prove this formula is correct we again take one of the points and see if the equation holds.
\n" ); document.write( "take (-1/4,1/2)
\n" ); document.write( "x = -1/4
\n" ); document.write( "y = 1/2
\n" ); document.write( "equation becomes -12*(-1/4) + 16*(1/2) = 11
\n" ); document.write( "this becomes 3 + 8 = 11
\n" ); document.write( "this becomes 11 = 11 proving the formula is correct.
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