document.write( "Question 696105: determine the slope and y intercept of the problems:y=5x+3, y=-8x-1,y=4+2x,(8,4) and (2,1), (-9,8) and (10,- 4), (-1,-2) and (-3,-4) \n" ); document.write( "

Algebra.Com's Answer #428840 by KMST(5328) $\"\"$ $\"About$

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If you have the equation for a line in the form $\"y=mx%2Bb\"$ ,
\n" ); document.write( "called the $\"highlight%28slope-intercept%29\"$ form
\n" ); document.write( "(or if you can transform the equation you have into that form),
\n" ); document.write( "the numbers $\"m\"$ and $\"b\"$ are the slope and y-intercept respectively.
\n" ); document.write( "For $\"y=5x%2B3\"$ , $\"highlight%28slope=5%29\"$ and $\"highlight%28y-intercept=3%29\"$
\n" ); document.write( "For $\"y=-8x-1\"$ , $\"highlight%28slope=-8%29\"$ and $\"highlight%28y-intercept=-1%29\"$
\n" ); document.write( "For $\"y=4%2B2x\"$ <--> $\"y=2x%2B4\"$ , $\"highlight%28slope=2%29\"$ and $\"highlight%28y-intercept=4%29\"$
\n" ); document.write( "
\n" ); document.write( "If you have two points, you can determine the slope and, the y-intercept, and the equation of the line.
\n" ); document.write( "I see two simple ways about it:
\n" ); document.write( "solve it as a system of equations,
\n" ); document.write( "or use what you learned about analytical geometry.
\n" ); document.write( "The choice of method depends on your preference (what is easier for you) and your teacher's preference.
\n" ); document.write( "
\n" ); document.write( "USING SYSTEMS OF EQUATIONS:
\n" ); document.write( "For the line that passes through (8,4) and (2,1),
\n" ); document.write( "substituting $\"x=8\"$ and $\"y=4\"$ (coordinates of (8,4)) into $\"y=mx%2Bb\"$ we get
\n" ); document.write( " $\"4=8m%2Bb\"$
\n" ); document.write( "Doing the same with point (2,1), we get
\n" ); document.write( " $\"1=2m%2Bb\"$
\n" ); document.write( "That gives you the system $\"system%284=8m%2Bb%2C2x%2Bb%29\"$
\n" ); document.write( "that you can solve for $\"m\"$ and $\"b\"$ .
\n" ); document.write( "Subtracting the second equation from the first, you get
\n" ); document.write( " $\"3=6m\"$ --> $\"m=3%2F6=1%2F2\"$ so $\"highlight%28slope=1%2F2%29\"$
\n" ); document.write( "Substituting that value into $\"1=2m%2Bb\"$ , you get
\n" ); document.write( " $\"1=2%281%2F2%29%2Bb\"$ --> $\"1=1%2Bb\"$ --> $\"b=0\"$ so $\"highlight%28y-intercept=0%29\"$
\n" ); document.write( "
\n" ); document.write( "USING ANALYTICAL GEOMETRY:
\n" ); document.write( "The slope of a line is defined as the ratio between the increase in y-coordinate and the increase in x-coordinate when going from one point to the other.
\n" ); document.write( "That is easier to write in words than as the usual cumbersome formula, which looks like
\n" ); document.write( " $\"slope=highlight%28m=%28y%5B1%5D-y%5B2%5D%29%2F%28x%5B1%5D-x%5B2%5D%29%29\"$
\n" ); document.write( "for known points $\"P%5B1%5D\"$ ( $\"x%5B1%5D\"$ , $\"y%5B1%5D\"$ ) and $\"P%5B1%5D\"$ ( $\"x%5B1%5D\"$ , $\"y%5B1%5D\"$ ).
\n" ); document.write( "Then, for any point (x,y) on the line, and a known point ( $\"x%5BA%5D\"$ , $\"y%5BA%5D\"$ )
\n" ); document.write( " $\"m=%28y-y%5BA%5D%29%2F%28x-x%5BA%5D%29\"$ --> $\"highlight%28y-y%5BA%5D=m%28x-x%5BA%5D%29%29\"$
\n" ); document.write( "gives the equation of the line in the $\"highlight%28point-slope%29\"$ form,
\n" ); document.write( "which can be transformed into the $\"highlight%28slope-intercept%29\"$ form.
\n" ); document.write( "
\n" ); document.write( "For the line that passes through (8,4) and (2,1),
\n" ); document.write( "so as not to get all those numbers mixed up,
\n" ); document.write( "you could write into the formula the coordinates of one point at a time.
\n" ); document.write( "You may start with point (8,4), with $\"x=8\"$ and $\"y=4\"$ , and write
\n" ); document.write( " $\"slope=%284-y%5B2%5D%29%2F%288-x%5B2%5D%29\"$ (leaving the spaces for $\"x%5B2%5D\"$ and $\"y%5B2%5D\"$ blank).
\n" ); document.write( "Then, you could fill in the coordinates of the other point to get
\n" ); document.write( " $\"slope=%284-1%29%2F%288-2%29=3%2F6=1%2F2\"$ so $\"highlight%28slope=1%2F2%29\"$
\n" ); document.write( "Then you can write the equation of the line in point-slope form,
\n" ); document.write( "maybe using point (2,1) as your point:
\n" ); document.write( " $\"y-1=%281%2F2%29%28x-2%29\"$
\n" ); document.write( "That can be transformed int the slope intercept form:
\n" ); document.write( " $\"y-1=%281%2F2%29%28x-2%29\"$ --> $\"y-1=%281%2F2%29x-1%29\"$ --> $\"highlight%28%281%2F2%29x%29\"$
\n" ); document.write( "The y-intercept is that invisible $\"0\"$ added after the $\"%281%2F2%29x\"$
\n" ); document.write( "so $\"highlight%28y-intercept=0%29\"$ .
\n" ); document.write( "
\n" ); document.write( "For the line that passes through (-9,8) and (10,- 4)
\n" ); document.write( "Slope= $\"%288-%28-4%29%29%2F%28-9-10%29=12%2F%28-19%29=highlight%28-12%2F19%29\"$
\n" ); document.write( " $\"y-%28-4%29=%28-12%2F19%29%28x-10%29\"$ --> $\"y%2B4=%28-12%2F19%29x%2B120%2F19\"$ --> $\"y=%28-12%2F19%29x%2B120%2F19-4\"$ --> $\"y=%28-12%2F19%29x%2B120%2F19-76%2F19\"$ --> $\"y=%28-12%2F19%29x%2B44%2F19\"$
\n" ); document.write( "So $\"highlight%28y-intercept=44%2F19%29\"$ .
\n" ); document.write( "
\n" ); document.write( "For the line that passes through (-1,-2) and (-3,-4)
\n" ); document.write( "Slope= $\"%28-2-%28-4%29%29%2F%28-1-%28-3%29%29=%28-2%2B4%29%2F%28-1%2B3%29=2%2F2=highlight%281%29\"$
\n" ); document.write( " $\"y-%28-4%29=1%28x-%28-3%29%29\"$ --> $\"y%2B4=x%2B3\"$ --> $\"y=x%2B3-4\"$ --> $\"y=x-1\"$
\n" ); document.write( "So $\"highlight%28y-intercept=-1%29\"$ . \n" ); document.write( "

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