Linear equations in one variable

Q: How do you solve an equation with brackets?

Expand the brackets first, then solve as usual. For 2(x + 3) = 14: x + 3 = 7, so x = 4.

A linear equation in one variable says that two expressions are equal and contains one unknown, like $3 x + 5 = 20$ . To solve it, do the same operation to both sides until the unknown stands alone.

Formulas

The idea: whatever you do to one side, do to the other

a x + b = c \Rightarrow a x = c - b \Rightarrow x = \frac{c - b}{a}

Variable on both sides: collect the x terms on one side

p x + q = r x + s \Rightarrow (p - r) x = s - q

Worked examples

Example 1. Solve

3 x + 5 = 20

Subtract 5 from both sides: $3 x = 15$ .
Divide both sides by 3: $x = 5$ .
Check: $3 \times 5 + 5 = 20$ . ✓

Answer:

x = 5

Example 2. Solve

5 x - 7 = 2 x + 11

Subtract $2 x$ from both sides: $3 x - 7 = 11$ .
Add 7 to both sides: $3 x = 18$ .
Divide by 3: $x = 6$ .

Answer:

x = 6

Example 3. A number is multiplied by 4 and then 9 is added. The result is 41. Find the number.

Let the number be $n$ . Then $4 n + 9 = 41$ .
$4 n = 32, so n = 8$ .

Answer: 8

Example 4. Solve

\frac{x}{3} + 2 = 7

Subtract 2: $\frac{x}{3} = 5$ .
Multiply both sides by 3: $x = 15$ .

Answer:

x = 15

Common mistakes

Changing only one side. Every step must keep the two sides equal.
Sign errors when moving a term across the equals sign. Moving a term changes its sign: $+ 5$ becomes $- 5$ .
Not checking the answer. Substitute back into the original equation to be sure.

Practice questions with answers

1. Solve for

x : x + 5 = 12

Show answer

Answer: B) 7

Subtract 5 from both sides:

x = 12 - 5 = 7

2. Solve

x + 9 = 16

Show answer

Answer: A) 7

Subtract 9 from both sides:

x = 16 - 9 = 7

3. Solve for

m : \frac{m}{5} = 4

Show answer

Answer: A) 20

Multiply both sides by 5:

m = 4 \times 5 = 20

4. Solve:

5 x = 3 x + 8

A $x = 4$
B $x = 2$
C $x = 8$
D $x = 1$

Show answer

Answer: A)

x = 4

5 x - 3 x = 8, so 2 x = 8, x = 4

5. A number increased by 7 gives 19. Find the number.

Show answer

Answer: D) 12

Let the number be

x . x + 7 = 19

, so

x = 19 - 7 = 12

6. What is

x

in the equation

5 x + 8 = 3 x + 8

Show answer

Answer: A) 0

Subtract

3 x : 2 x + 8 = 8

. Subtract 8:

2 x = 0

. So

x = 0

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Frequently asked questions

What is a linear equation in one variable?

An equation with one unknown whose highest power is 1, such as $2 x - 3 = 9$ .

How do you solve an equation with brackets?

Expand the brackets first, then solve as usual. For $2 (x + 3) = 14 : x + 3 = 7$ , so $x = 4$ .

How can I check my answer?

Put the value back into the original equation. If both sides are equal, the answer is correct.

How do you turn a word problem into an equation?

Let a letter stand for the unknown, write each statement in the problem as an expression, then set the two equal expressions equal to each other.

Linear equations in one variable

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

Related topics