SAT/ACT: Lines and Slopes

2. Slope of a line can also be zero or not defined.

If a line is parallel to the x-axis, then the line has zero slope. A line parallel to the x-axis means that there is no change in the y-axis as the x-axis value changes.

If a line is parallel to the y-axis, then the line has an undefined slope. A line parallel to the y-axis means that there is no change in the x-axis as the y-axis value changes.

3. The steeper the line, the higher the slope.

If a line looks difficult to walk up, then that line has a steeper slope.

For example, in the figure above, it is easier to move up the yellow line and more difficult to move up the red line. Hence, the red line has a steeper slope.

4. Two parallel lines have the same slope.

As you can see from the figure above, it is equally difficult to move up both the lines, hence, the two lines have the same slope.

5. Two perpendicular lines have slopes that are negative inverse of each other.

Typical questions on the SAT or the ACT

Q1. Which of the following is a line that is perpendicular to the line y = 2x + 3?

1. 2y = x + 5
2. 2y = -x + 5
3. 2y = -x/2 + 5
4. 2y = -2x + 5

Solution:

For two lines to be perpendicular, their slopes must be negative inverse of each other.

For the original equation, the slope is 2.

Hence, the slope of a line perpendicular to this line, the new slope must be -½
​​​​

1. 2y = x + 5

=> y = x/2 + 5/2………….Nope….the slope here is 1/2

2. 2y = -x + 5

=> y = -x/2 + 5/2………..Bingo…..the slope is -½

3. 2y = -x/2 + 5

=> y = -x/4 + 5/2……...Nah……...the slope is -¼

4. 2y = -2x + 5

=> y = -x + 5/2………..No……….the slope of -1

Hence, the answer is (B).

Always write your equations in the general form.

Q2.

In the xy-plane, a line passes through points (-3, 5) and (6, 8). Which of the following points is also on this line?

1. (0, 6)
2. (3, 8)
3. (9, 10)
4. (12, 11)

Solution:

Slope of the given line = (y2 - y1)/(x2 - x1) = (8 - 5)/(6 - (-3)) = 3/9 = 1/3

For any other point to be on the line, it must have the same slope with the given points.

1. (0, 6) and (6, 8)

Slope = (8 - 6)/(6 - 0) = 2/6 = 1/3

2. (3, 8) and (6, 8)

Slope = (8 - 8)/(6 - 3) = 0

3. (9, 10) and (6, 8)

Slope = (8 - 10)/(6 - 9) = -2/-3 = 2/3

4. (12, 11) and (6, 8)

Slope = (8 - 11)/(6 - 12) = -3/-6 = 1/2

Only answer choice (A) gives the same slope, hence, that’s the point on the line.

Q3.  

Equation of line l, shown above, is y = mx + c. Which of the following must be true based on the information provided and the figure?

1. mc > 0
2. mc < 0
3. mc = 0
4. mc = 1

Solution:

As the line moves up from left to right, the slope is positive, hence, m > 0

The line cuts the y-axis on the negative y, hence, c < 0.

mc = (+ve)(-ve) = -ve < 0

So, the correct answer is (B).

Q4. 

If a, b, c, and d represent the value of slopes of the different lines, which of the following must be true?

1. a < b < c < d
2. a < c < b < d
3. c < a < d < b
4. c < d < a < b

Solution:

c is moving down from left to right, hence, c must be negative.

Eliminate answer choices (A) and (B).

Between a and d, d is steeper, hence, d > a

Eliminate answer choice (D).

So, the best answer is (C).

These are some of the most common question types. Keep practicing and improving!

Happy prepping!!