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Linear equations

This page is part of the section “Task 1 USE profile level. Elementary equations”. Here are solutions to linear equations with neat integer answers. Each problem includes the condition, a step-by-step solution, and the answer.

Problems with solutions

Problem 1

Solve the equation:

3x−7=2x+5

Move the terms with x to the left side and the numbers to the right side, changing the sign when moving:

3x−2x=5+7 ⇒ x=12

Answer: x=12.

Problem 2

Solve the equation:

5(x−3)=2x+9

Expand the brackets on the left side, then move the terms with x to the left and the numbers to the right:

5x−15=2x+9 ⇒ 3x=24 ⇒ x=8

Answer: x=8.

Problem 3

Solve the equation:

x+43 = 2x−15

Multiply both sides by 15 (the least common multiple of the denominators 3 and 5) to eliminate the fractions:

5(x+4)= 3(2x−1) 5x+20=6x−3 ⇒ x=23

Answer: x=23.

Problem 4

Solve the equation:

0.4x+1.2=0.8x−2.4

Move the terms with x to the left side and the numbers to the right side:

0.4x−0.8x=−2.4−1.2 ⇒ −0.4x=−3.6 ⇒ x=9

Answer: x=9.

Frequently asked questions

How to solve linear equations from Task 1 of the USE?

Expand the brackets (if any), eliminate fractions (by multiplying both sides by the common denominator), move the terms with x to one side and the numbers to the other, combine like terms and find x.

What to do if the equation contains fractions?

Multiply both sides of the equation by the least common multiple of the denominators. Then the fractions disappear, and the equation reduces to a linear one without fractions.

Why does the sign change when a term is moved?

Because moving a term means adding the opposite number to both sides of the equation. The equality is preserved, and the sign of the moved term changes to the opposite.

How to check the answer of a linear equation?

Substitute the found value of x into the original equation and make sure that the left and right sides are equal. This is especially useful when working with fractions and decimal coefficients.

See also