Answer:
[tex]r=6[/tex]
Step-by-step explanation:
Make an equation in point-slope form where:
Insert the known values:
[tex](-2_{x1},8_{y1})\\\\m=-\frac{1}{2} \\\\y-8=-\frac{1}{2} (x-(-2))\\\\y-8=-\frac{1}{2} (x+2)[/tex]
Use the distributive property:
[tex]-\frac{1}{2} (x+2)\\\\-\frac{1}{2}(x)-\frac{1}{2}(2)[/tex]
Simplify. Turn the 2 into a fraction by converting to the fraction [tex]\frac{2}{1}[/tex] , which is still equal to 2:
[tex]-\frac{1}{2}(2)\\\\-\frac{1}{2}*\frac{2}{1}[/tex]
Multiply across:
[tex]-\frac{1}{2}*\frac{2}{1} =-\frac{2}{2} =-1[/tex]
Re-insert into the equation:
[tex]y-8=-\frac{1}{2}x-1[/tex]
Solve for y. Add 8 to both sides (to isolate y using inverse operations):
[tex]y-8+8=-\frac{1}{2}x-1+8\\\\y=-\frac{1}{2}x+7[/tex]
Now insert the y value of the other point to find r (which will have the same value as x):
[tex](r_{x},4_{y})\\\\4=-\frac{1}{2}x+7[/tex]
Simplify the fraction by multiplying. Add 1 as the denominator for x:
[tex]-\frac{1}{2}*\frac{x}{1}[/tex]
Multiply across:
[tex]-\frac{1}{2}*\frac{x}{1}=-\frac{x}{2}[/tex]
Re-insert:
[tex]4=-\frac{x}{2} +7[/tex]
Subtract 7 from both sides (to isolate the variable using inverse operations):
[tex]4-7=-\frac{x}{2} +7-7\\\\-3=-\frac{x}{2}[/tex]
Multiply both sides by 2 (to undo the fraction using inverse operations):
[tex]2(-3)=2(-\frac{x}{2}) \\\\-6=-x[/tex]
Divide both sides by -1 (to make the variable positive, since two negatives make a positive):
[tex]\frac{-6}{-1} =\frac{-x}{-1} \\\\x=6[/tex]
Therefore, the value of r is 6.
:Done
