Which of the following is the equation of the line that passes through the points and ?
(A)
(B)
(C)
(D)
(E)
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Correct Answer: E
Solution 1:
Tip: The slope of a line is defined as change in x change in y .
Option (E) says that the slope of the line that passes through points ( x , y ) and ( 1 , 3 ) is equal to the slope of the line that passes through the points ( x , y ) and ( − 2 , 5 ) . Thus, this point must lie on the line between ( 1 , 3 ) and ( − 2 , 5 ) , which gives us the answer.
Solution 2:
Tip: The slope of a line is defined as change in x change in y .
Tip: Point-slope form: y − y 1 = m ( x − x 1 ) , where m is the line's slope, and ( x 1 , y 1 ) is a point on the line.
Let's find the equation of the line. The slope of the line is equal to
m = change in x change in y = − 2 − 1 5 − 3 = − 3 2 .
By the point-slope form, the equation of this line is
( y − 3 ) 3 ( y − 3 ) 3 y − 9 3 y + 2 x − 1 1 = = = = − 3 2 ( x − 1 ) − 2 ( x − 1 ) − 2 x + 2 0 point-slope form multiply both sides by 3 expand terms simplify
Now, we check this equation against the answer choices.
(A) y = − 2 x + 5 is not the same as 3 y + 2 x − 1 1 = 0 .
(B) y = 2 x + 1 is not the same as 3 y + 2 x − 1 1 = 0 .
(C) If x − 1 y − 3 = 3 2 , then cross multiplying gives us 3 ( y − 3 ) = 2 ( x − 1 ) . Expanding gives 3 y − 9 = 2 x − 2 . Simplifying gives 3 y − 2 x − 7 = 0 . This is not the same as 3 y + 2 x − 1 1 = 0 .
(D) If x + 2 y − 5 = − 2 3 , then cross multiplying gives 2 ( y − 5 ) = − 3 ( x + 2 ) . Expanding gives 2 y − 1 0 = − 3 x − 6 . Simplifying gives 2 y + 3 x − 4 = 0 . This is not the same as 3 y + 2 x − 1 1 = 0 .
(E) If x − 1 y − 3 = x + 2 y − 5 , then cross multiplying gives ( x + 2 ) ( y − 3 ) = ( x − 1 ) ( y − 5 ) Expanding gives x y + 2 y − 3 x − 6 = x y − y − 5 x + 5 . Simplifying gives 3 y + 2 x − 1 1 = 0 . Hence, this is the answer.
Solution 3:
Tip: Plug and check.
Each of the two points, when plugged into the equation of the line that passes through them should yield a true statement. So we plug and check. Here, for each wrong answer, we only dothe work for the point that yields a contradiction.
(A) − 2 × ( − 2 ) + 5 = 9 = 5 so ( − 2 , 5 ) does not lie on the line.
(B) 2 × ( − 2 ) + 5 = 1 = 5 so ( − 2 , 5 ) does not lie on the line.
(C) − 2 − 1 5 − 3 = − 3 2 = 3 2 , so ( − 2 , 5 ) does not lie on the line.
(D) 1 + 2 3 − 5 = 3 − 2 = − 2 3 , so ( 1 , 3 ) does not lie on the line.
We are left with option (E), which has to be the answer.
Incorrect Choices:
(A) , (B) , (C) and (D)
Look at Solution 3 for how to eliminate these choices by plugging and checking.