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IntermediateGrades 6–716 min16 / 82

Linear equations

The root of an equation, the key properties of equations, solving ax + b = 0 step by step and solving word problems with equations.

Check yourself
In this lesson you will learn
  • Find the root of an equation and check it
  • Use the basic properties of equations
  • Solve linear equations with brackets and fractions
  • Set up an equation from a word problem

Elvin thought of a number, multiplied it by 3, added 7 and got 25. What number was he thinking of? The puzzle can be written briefly as 3x + 7 = 25. This is an equation — once you learn how to solve it, you can crack every puzzle of this kind in a few steps.

An equation and its root

Definition
Equation

An equality that contains an unknown number (usually written as x). A value of the unknown that makes the equality true is called a root (or solution) of the equation. To solve an equation means to find all its roots or to show that it has none.

Imagine an equation as a balanced pair of scales. If you add the same weight to both pans or take the same weight away from both, the balance is kept. The basic properties of equations work the same way:

  1. You may add the same number to both sides or subtract the same number from both sides. That is why a term can be moved to the other side of the equation if you change its sign.
  2. You may multiply or divide both sides by the same non-zero number.
ax + b = 0 ⇒ x = −b/a (a ≠ 0)ax + b = 0 ⇒ x = −b/a (a ≠ 0)
where:
  • xthe unknown
  • athe coefficient of x
  • bthe constant term

Such an equation is called linear: the unknown appears only to the first power.

Elvin's puzzle

Solve 3x + 7 = 25.

Show solution
Move 7 to the right side, changing its sign: 3x = 25 − 7.
3x = 18.
Divide both sides by 3: x = 6.
Check: 3 · 6 + 7 = 18 + 7 = 25 — correct.

A step-by-step method

  1. 1
    Expand brackets, clear fractions

    If there are fractions, multiply both sides by the LCM of the denominators.

  2. 2
    Collect the terms

    Move the x terms to the left and the numbers to the right; a term changes its sign when it crosses the equals sign.

  3. 3
    Combine like terms

    The equation becomes ax = c.

  4. 4
    Divide by the coefficient and check

    x = c/a. Substitute the answer into the original equation to check it.

An equation with brackets

Solve 5(x − 2) = 3x + 4.

Show solution
Expand the brackets: 5x − 10 = 3x + 4.
x terms to the left, numbers to the right: 5x − 3x = 4 + 10.
2x = 14, x = 7.
Check: 5(7 − 2) = 25 and 3 · 7 + 4 = 25.
An equation with fractions

Solve x/2 + x/3 = 10.

Show solution
LCM(2, 3) = 6. Multiply both sides by 6: 3x + 2x = 60.
5x = 60, x = 12.
Check: 12/2 + 12/3 = 6 + 4 = 10.

Solving word problems with equations

Books

Aysel and Leyla have 34 books altogether. Leyla has 6 more books than Aysel. How many books does each girl have?

Show solution
Let x be the number of Aysel's books. Then Leyla has x + 6.
Equation: x + (x + 6) = 34.
2x + 6 = 34, 2x = 28, x = 14.
Aysel has 14 books and Leyla has 14 + 6 = 20. Check: 14 + 20 = 34.

Key points

  • A root of an equation is a number that makes it a true equality.
  • A term changes its sign when moved to the other side.
  • Both sides can be multiplied or divided by the same non-zero number.
  • ax + b = 0 (a ≠ 0) has exactly one root, x = −b/a.
  • Always check your answer in the original equation.

Check yourself

10 questions. Every correct answer earns XP.

1 / 10
Which number is a root of 2x − 3 = 7?