Tayarri LCAT
LCAT Math, Algebra

Linear equations for the LCAT: lesson and practice

To solve a linear equation on the LCAT, simplify each side, move the x-terms to one side and the numbers to the other, then divide by the coefficient of x. The LCAT also asks when an equation has no solution (same x-coefficient, different constants) or infinitely many (both sides identical).

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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.

How to solve a linear equation, step by step

  1. Clear brackets. Multiply the outside number by every term inside.
  2. Clear fractions by multiplying every term by the common denominator.
  3. Collect x-terms on one side and numbers on the other, doing the opposite operation each time something crosses the equals sign.
  4. Divide by the number in front of x.
  5. Check by putting your answer back into the original equation.

Example 1: brackets on one side

If 3(x − 4) = 2x + 5, what is x?

3x − 12 = 2x + 5
x − 12 = 5
x = 17

Check: 3(17 − 4) = 39 and 2(17) + 5 = 39. Both sides match.

Example 2: a word problem

A ride costs a booking fee of Rs 100 plus Rs 30 per kilometer. The total was Rs 400. How long was the ride?

100 + 30k = 400
30k = 300
k = 10 km

The fixed fee comes off before dividing. Dividing 400 by 30 is the most common trap.

No solution and infinitely many solutions

These are among the most-missed linear equation questions on the LCAT. Simplify both sides fully, then compare.

After simplifyingExampleNumber of solutions
Different x-coefficients5x + 2 = 3x + 8Exactly one
Same x-coefficient, different constants6x + 8 = 6x + 5None
Both sides identical4x + 10 = 4x + 10Infinitely many

Example 3: find k for no solution

In 2(3x + 4) = kx + 5, for what value of k is there no solution?

6x + 8 = kx + 5

The constants (8 and 5) already differ, so we need equal x-coefficients: k = 6. Think of two parallel lines: same slope, never meeting.

Common mistakes

The practice above tracks these for you. When you make one twice in a row, you get a targeted lesson and a new set of questions.

Not distributing to every term3(x − 4) is 3x − 12, not 3x − 4.
Sign errors when moving terms+5 on one side becomes −5 on the other.
Stopping one step early4x = 24 is not the answer. Divide: x = 6.
Ignoring a fixed amountIn "fee + rate × distance" problems, remove the fee before dividing.
Answering for the wrong quantityIf the question asks for x + y or the largest integer, finding x is not the end.
Adding fractions by adding denominatorsx/2 + x/3 is 5x/6, not x/5.

Sample questions with answers

1. If 3(x − 4) = 2x + 5, what is the value of x?
  1. 9
  2. 17 (correct)
  3. −7
  4. 1

Distribute: 3x − 12 = 2x + 5. Subtract 2x and add 12: x = 17.

2. In the equation 2(3x + 4) = kx + 5, k is a constant. For what value of k does the equation have no solution?
  1. 3
  2. 6 (correct)
  3. 8
  4. −6

Distribute: 6x + 8 = kx + 5. No solution means equal x-coefficients with different constants, so k = 6.

3. A ride costs a booking fee of Rs 100 plus Rs 30 per kilometer. The total was Rs 400. How many kilometers was the ride?
  1. 13
  2. 10 (correct)
  3. 300
  4. 17

100 + 30k = 400, so 30k = 300 and k = 10.

Frequently asked questions

When does a linear equation have no solution?

A linear equation has no solution when, after simplifying, both sides have the same x-coefficient but different constants, such as 6x + 8 = 6x + 5. It has infinitely many solutions when both sides are identical.

What is the most common mistake with linear equations on the LCAT?

Not distributing to every term inside brackets, such as writing 3(x − 4) as 3x − 4 instead of 3x − 12, and sign errors when moving terms across the equals sign.

Is linear equations 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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