BackClose

Select a Subject

Subject Test Math 1

Question 8 of 32

beginning of content:
Tags:
SAT Subject Test

In the x y text -plane end text, the points with coordinates left parenthesis 0 comma negative 5 right parenthesis and left parenthesis 6 comma negative 2 right parenthesis lie on line l. Line p contains the point with coordinates left parenthesis negative 5 comma 0 right parenthesisand is perpendicular to line l. What is the x text -coordinate end text of the point where lines l and p intersect?

Select an Answer

negative 6

Correct Answer: 
No

negative 5

Correct Answer: 
No

negative 4

Correct Answer: 
No

negative 3

Correct Answer: 
No

negative 2

Correct Answer: 
Yes

It is often helpful to make a sketch for this type of question.

The figure presents two lines, l and p, in the x y-plane with the origin labeled O. The number 1 is indicated on the x-axis. The number negative 1 is indicated on the y-axis. Line l passes through the points with coordinates 0 comma negative 5 and 6 comma negative 2. Line p passes through the point with coordinates negative 5 comma 0. The lines intersect at the point with coordinates negative 2 comma negative 6.

The slope of line l is equal to fraction numerator negative 5 invisible times minus left parenthesis negative 2 right parenthesis over denominator 0 invisible times minus 6 end fraction equals fraction numerator negative 3 over denominator negative 6 end fraction invisible times equals fraction numerator 1 over denominator 2 end fraction.
Since negative 5 is the y text -intercept end text, the equation of l is y invisible times equals fraction numerator 1 over denominator 2 end fraction x invisible times minus 5.

Since line p is perpendicular to line l, the slope of p is equal to negative 2 (the negative reciprocal of the slope of l ). Since line p contains the point with coordinatesleft parenthesis negative 5 comma 0 right parenthesis, the equation of line p can be found by substitution.

number space number space number space table attributes columnalign right center left columnspacing 2px end attributes row y equals cell m x plus b end cell row cell       0 end cell equals cell left parenthesis negative 2 right parenthesis left parenthesis negative 5 right parenthesis plus b end cell end table
table attributes columnalign right center left columnspacing 2px end attributes row cell back space number space minus 10 end cell equals b row y equals cell negative 2 x minus 10 end cell end table

To find the x text -coordinate end text of the point where the two lines intersect, solve the equation fraction numerator 1 over denominator 2 end fraction x invisible times minus 5 invisible times equals negative 2 x invisible times minus 10. The lines intersect at the point where x invisible times equals negative 2.

It is also possible to graph the two linear equations to determine where they intersect. Using a graphing calculator, let y subscript 1 end subscript invisible times equals fraction numerator 1 over denominator 2 end fraction x invisible times minus 5 and y subscript 2 end subscript invisible times equals negative 2 x invisible times minus 10. You can find the point of intersection graphically in the standard viewing window, or you can examine a table of values to find the x text -value end text where y subscript 1 end subscript invisible times equals y subscript 2 end subscript.

Question Difficulty: 
medium