Tayarri LCAT
LCAT Math, Algebra

Linear functions for the LCAT: lesson and practice

A linear function has the form f(x) = mx + b. The slope m is how much f(x) changes each time x goes up by 1, and the intercept b is the value when x = 0. LCAT questions ask you to find the slope, write the equation of a line, explain what the slope and intercept mean in a real situation, and use parallel and perpendicular slopes.

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No calculator on the LCAT. LUMS’s official guidelines say calculators are not allowed, so every practice question here can be solved by hand.

Finding the slope

Slope is the change in y divided by the change in x. Subtract in the same order on the top and the bottom.

slope = (y₂ − y₁) ÷ (x₂ − x₁)
Points (1, 2) and (4, 11): (11 − 2) ÷ (4 − 1) = 9 ÷ 3 = 3
You are givenHow to get the slope
Two points(y₂ − y₁) ÷ (x₂ − x₁)
y = mx + bThe number in front of x
ax + by = c−a/b (or solve for y)
A tableChange in f(x) ÷ change in x between any two rows

What slope and intercept mean in word problems

The LCAT often gives a model like C = 30d + 100 and asks what a number means.

Example: a ride-hailing fare

C = 30d + 100 gives the cost in rupees of a ride d kilometers long.

30 is multiplied by the variable, so it is the rate: Rs 30 for each kilometer. 100 stands alone, so it is the starting value: the booking fee paid even for a 0-kilometer ride.

Quick test: set the variable to 0. Whatever remains is the intercept. The number that changes the total for each extra unit is the slope.

Writing the equation of a line

  1. Find the slope from two points.
  2. Substitute one point into y = mx + b to find b.
  3. Check with the other point. Both points must work.

Example

Line through (2, 7) and (5, 16).

slope = (16 − 7) ÷ (5 − 2) = 3
7 = 3(2) + b, so b = 1
y = 3x + 1

Parallel and perpendicular lines

RelationshipSlopesExample
ParallelEqual2/3 and 2/3
PerpendicularNegative reciprocals (product is −1)2/3 and −3/2

Common mistakes

Putting x on topSlope is rise over run: change in y on top.
Subtracting in different orders(y₂ − y₁) must pair with (x₂ − x₁), or the sign flips.
Swapping slope and interceptIn C = 30d + 100, the fee is 100, not 30.
Using a point's y-value as the interceptThe intercept is the y-value only when x = 0.
Perpendicular slope without the sign changeThe perpendicular to 2/3 is −3/2, not 3/2.

Sample questions with answers

1. A line passes through (1, 2) and (4, 11). What is its slope?
  1. 1/3
  2. 3 (correct)
  3. −3
  4. 11/4

(11 − 2) ÷ (4 − 1) = 3. Choice A puts the change in x on top.

2. C = 30d + 100 gives the cost in rupees of a ride d kilometers long. What does 100 represent?
  1. The cost per kilometer
  2. The booking fee (correct)
  3. The cost of a 1-kilometer ride
  4. The length of a ride

100 is the cost when d = 0, the fixed booking fee. 30 is the cost per kilometer.

3. Line k has slope 2/3. What is the slope of a line perpendicular to k?
  1. 2/3
  2. 3/2
  3. −3/2 (correct)
  4. −2/3

Perpendicular slopes are negative reciprocals: flip 2/3 to 3/2 and change the sign.

Frequently asked questions

What is the slope formula for the LCAT?

Slope equals the change in y divided by the change in x: (y2 − y1) ÷ (x2 − x1). Subtract in the same order on the top and bottom.

How do you find the slope from standard form?

For an equation written as ax + by = c, the slope is −a/b. You can also solve the equation for y and read the number in front of x.

What does the y-intercept mean in a word problem?

It is the value of the output when the input is 0, usually a starting amount or a fixed fee.

What is the slope of a perpendicular line?

The negative reciprocal. If one line has slope 2/3, a perpendicular line has slope −3/2. The two slopes multiply to −1.

Is linear functions tested on the LCAT?

Yes. It falls under Algebra, one of the content domains LUMS lists for the LCAT in its official sample questions.

Can I practice this without a calculator?

Yes. Every practice question here can be solved by hand, because calculators are not allowed on the LCAT.

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