SAT/ACT: Lines and Slopes

Remember your xy-plane? Yes, the same graph where the x- and they-axis meet perpendicular to each other. Any point on this plane is given as (x, y). We can draw many different types of shapes such as line, parabola, circle and many more on this plane. However, this article will talk only about a line.

A line is something that is straight and involves an infinite array of points in its path. So, how do we differentiate one line from another? We do so based on the slope of the line. Slope informs us about the steepness of the line. A slope can be calculated as the change on the y-axis as the distance changes on the x-axis.

Slope = change in y-axis/change in x-axis

If a line passes through points (x1, y1) and (x2, y2), then the slope = (y2 - y1)/(x2 - x1).

How is this concept tested on the test?

To understand the application of the concept, you must first understand the equation of a line.

The general equation of a line is y = mx + c, where

y = the y-coordinate value for any x-coordinate

x = the x-coordinate value for any y-coordinate

m = slope or gradient

c = y-intercept (the point where the line cuts the y-axis)

You must always write your equation in the general form to avoid falling for any traps.

Example 1:

mx + 5y = 3

The equation above is a straight line in the xy-plane. If the slope of line is 3, what is the value of m?

Solution:

First change the equation of the line in the general equation form.

=> 5y = -mx + 3

y = (-m/5)x + ⅗

Now, slope is the coefficient of x.

=> (-m/5) = 3

=> -m = 15

or m = -15

Some properties to keep in mind:

1. Slope of a line can be positive or negative.

Positive slopes move up from left to right and

Negative slopes move down from left to right.

In a positive slope, the value of y increases as the value of x increases. For example, the amount of money you make increases as the number of hours you put in a work increases.

In a negative slope, the value of y decreases as the value of x increases. For example, the amount of money you are left with as the number of hours you put in for your shopping.

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!!