ANSWER
[tex] \boxed {y = - 2x + 7}[/tex]
EXPLANATION
We want to write the equation in slope-intercept form for the line that passes through the point (3,2) and the intersection of the lines:
[tex]2x - 3y = 24...(1)[/tex]
[tex]2x + y = 8...(2)[/tex]
We subtract equation (1) from equation (2) to get,
[tex]y - - 3y = 8 - 24[/tex]
[tex]4y = - 16[/tex]
[tex]y = - 4[/tex]
Put y=-4 into equation (2) to get the value of x.
[tex]2x - 4 = 8[/tex]
[tex]2x = 8 + 4[/tex]
[tex]2x = 12[/tex]
[tex]x = 6[/tex]
Therefore the line passes through (3,2) and
(6,-4).
The slope of this line is
[tex]m = \frac{ - 4 - 2}{6 - 3} = \frac{ - 6}{3} = - 2[/tex]
The slope intercept form is given by the formula,
[tex]y = mx + c[/tex]
where m=-2 is the slope.
We substitute the slope to get,
[tex]y = - 2x + c[/tex]
We substitute the point (3,2) to find the value of c.
[tex]3 = - 2(2) + c[/tex]
[tex]3 = - 4 + c[/tex]
[tex]c = 4 + 3 = 7[/tex]
Hence the equation in slope-intercept form is
[tex]y = - 2x + 7[/tex]
