2.1 PSAT/SAT Algebra Practice. Graphing Linear Equations

Line A is represented by the line y=3x−4. Which of the following represents the equation of a line that is perpendicular to Line A and that travels through point (1,2)?

• y = −1/3x + 7/3

• y = −3x + 4

• y = −3x + 34

• y = −13x + 43

Answer explanation

This problem tests two key principles related to lines in the coordinate plane. For one, the slopes of perpendicular lines are negative reciprocals of each other (e.g. a and −1a.) So here since you need to find a line perpendicular to a line with slope of 3, you're looking for the negative reciprocal −1/3. You can therefore eliminate choices y=−3x+4 and y=−3x+34, which do not have the proper slope.

The second key concept relates to what a point in the coordinate plane is. Because points are all given in the form (x,y) you know that for a point to pass through the point (1,2) it will need to allow for x to equal 1 when y equals 2. To find the b term (in y=mx+b), take what you do know about the line:

y= −1/3x + b

From here, add 1/3 to each side to isolate b and you'll have your answer:

7/3=b, so the correct equation is y=−1/3x+7/3.

Which of the following equations represents a line that is perpendicular to y = 1/2x + 2?

• 3y+6x=12

• 4y−8x=16

• y−2x=8

• 2x+4y=10

Answer explanation

In order for two lines to be perpendicular, the slopes need to be negative reciprocals of each other. Since the given slope is 12, you're looking for a slope of −2.

To find the slope, you need to put the equations in point-slope form, y=mx+b, where m is the slope. Checking the answer choices quickly:

Choices y−2x=8 and 4y−8x=16 will each give x a positive coefficient, as your first step is to get y on the opposite side of the equation from x.

Only choice 3y+6x=12 simplifies to a coefficient of −2:

3y+6x=12

3y=−6x+12

y=−2x+4

Line J has a y-intercept of 6 and passes through point (a,b). What is the slope of Line J in terms of a and b?

• 6 − b

  a

• b - 6

  a

• a - b

  6

• 6 - a

  b

Answer explanation

ou know that the line goes through point (a,b) and the y-intercept of 6 tells you that the line also goes through point (0,6). So you can use the "rise over run" slope formula to calculate the slope. Recognizing that the five answer choices focus on a,b, and 6, you should see that you should subtract 0 and 6 (subtracting 0 is the same as doing nothing at all) to get your math looking like the answer choices. So you'll set up the equation:

b−6

a-0

In the coordinate plane, Line J passes through the origin as well as points (a,2) and (8,a). Which of the following could be the slope of Line J?

• 2

• 1

• 1/2

• 1/4

Answer explanation

That Line J passes through the origin gives you a very helpful bit of information regarding point-slope form (y=mx+b). That tells you that when x=0,y=0, meaning that 0=0+b⋯ meaning that b also equals 0. That takes a variable away, allowing you to get to work on the two points knowing that the line has a constant slope, m:

For point (a,2), that means that 2=ma.

For point (8,a), that means that a=8m.

This means that you can plug in a=8m into the first equation, allowing you to solve for m (the slope):

2 = 8m²

1/4=m²

m=1/2 (or −1/2).

Which of the following points does the line y = 3/2x + 4 NOT pass through?

• (2, 7)

• (14, 25)

• (-5, -3.5)

• (6, 12)

Answer explanation

In order to attack this question, we can plug in the value of x for each coordinate to see if the proper value of y matches. for 

(2, 7) if we plug in 2 for x, we can see that

y=3/2(2)+4

which does in fact = 7. 

For (-5, -3.5), if we take the same step we can see that 

y=3/2(−5)+4

does give us a y value of -3.5.

For (14, 25), plugging in x = 14 gets us to

y=3/2(14)+4

which simplifies to 25.

So, by default we can see that our correct answer must be 

(6, 12), since an input of 6 gives us 

y=3/2(6)+4

which equals 13, not 12. So, (6, 12) is the only point not on the line.