PETE C. answered • 10/07/20
Texas STAAR, Algebra, SAT MATH, GED
In my opinion this is a very poorly written question. It is ACTUALLY TWO SEPARATE ALGEBRA PROBLEMS.
Find k so that the line through (5,3) and (k,5) is a) parallel to 3y−2x=2; b) perpendicular to 2y+5x=7.
First problem, find K so that the line passes through (5,3) and (K,5)
AND IS PARALLEL to the line 3Y - 2X = 2.
This means the line has the SAME SLOPE as 3Y - 2X = 2.
3Y - 2X = 2
3Y = 2X + 2 Add 2X to both sides.
Y = (2/3)X + 2/3 Divide both sides by 3.
So, the SLOPE of the line that passes through the above 2 points is 2/3.
Now, plug in the slope and use the two points (5,3) and (K,5) to determine K.
m = RISE/RUN
2/3 = (5 - 3)/(K - 5)
2(K - 5) = 3(5 - 3) Cross Multiplication
2K - 10 = (3)(2)
2K - 10 = 6
2K = 6 + 10 Add 10 to both sides.
2K = 16
K = 16/2 Divide both sides by 2.
K = 8
ANSWER TO FIRST PROBLEM: K = 8
CHECKING USING (5,3) and (8,5):
m = (5-3) / (8 - 5) = 2 / 3 Yes!
Second problem, find K so that the line passes through (5,3) and (K,5)
and is PERPENDICULAR to line 2Y + 5X = 7.
Find the SLOPE of 2Y + 5X = 7.
2Y + 5X = 7
2Y = - 5X + 7 Subtract 5X from both sides.
Y = - (5/2)X + 7/2 Divide both sides by 2
So, the slope of the line 2Y + 5X = 7 is -(5/2).
Since the line containing the two points (5,3) and (K,5) is PERPENDICULAR to the line having a slope of -(5/2), the line containing the two points has a negative reciprocal of -(5/2) which is 2/5. In other words, when you multiply the two slopes, you get negative one.
-(5/2) (2/5) = - 10/10 = -1
So, the SLOPE of the line containing the 2 points is 2/5.
Now, plug in the slope and use the two points (5,3) and (K,5) to determine K.
m = RISE/RUN
2/5 = (5 - 3)/(K - 5)
2(K - 5) = 5(5 - 3) Cross Multiplication
2K - 10 = (5)(2)
2K - 10 = 10
2K = 20 Add 10 to both sides.
K = 20/2 Divide both sides by 2.
K = 10
ANSWER TO THE SECOND PROBLEM: K = 10
CHECKING USING (5,3) and (10,5):
m = (5-3) / (10 - 5) = 2 / 5 Yes!
Hope this helps you Brooke.
