Equations with Brackets & Fractions

Equations with Brackets

When solving linear equations with brackets, the key steps are to expand the brackets, collect like terms, and then isolate the variable. Always check your solution by substituting it back into the original equation.

Expanding brackets means multiplying each term inside the bracket by the term outside. For example, in $3(x + 4)$, multiply 3 by both $x$ and 4:

$$3(x + 4) = 3 \times x + 3 \times 4 = 3x + 12$$

After expanding, collect like terms on each side to simplify the equation. Then rearrange to isolate the variable on one side.

For instance, consider the equation:

$$2(x + 3) - 4 = 3x - 5$$

Expand the brackets:

$$2x + 6 - 4 = 3x - 5$$

Simplify the left side:

$$2x + 2 = 3x - 5$$

Collect like terms by subtracting $2x$ from both sides:

$$2 = x - 5$$

Add 5 to both sides to isolate $x$:

$$x = 7$$

Check by substituting $x = 7$ back into the original equation:

$$2(7 + 3) - 4 = 3 \times 7 - 5 \implies 2 \times 10 - 4 = 21 - 5 \implies 20 - 4 = 16 \implies 16 = 16$$

The solution is correct.

For example, expanding and simplifying $5(2x - 3)$ gives $10x - 15$.

Equations with Fractions

When equations contain fractions, the first step is to clear the denominators by multiplying both sides of the equation by the least common denominator (LCD). This removes the fractions and makes the equation easier to solve.

For example, solve:

$$\frac{x}{3} + 2 = 5$$

Multiply both sides by 3 (the denominator):

$$3 \times \frac{x}{3} + 3 \times 2 = 3 \times 5$$

$$x + 6 = 15$$

Subtract 6 from both sides:

$$x = 9$$

Always check for extraneous solutions by substituting your answer back into the original equation, especially when variables appear in denominators (to avoid division by zero). For example, if $x$ were in the denominator, check that your solution does not make the denominator zero.

Combined Brackets and Fractions

Equations may contain both brackets and fractions. The approach is to first clear fractions by multiplying both sides by the LCD, then expand brackets, collect like terms, and finally isolate the variable.

For example, solve:

$$\frac{3(x - 2)}{5} = \frac{2x + 1}{10}$$

The denominators are 5 and 10, so the LCD is 10. Multiply both sides by 10:

$$10 \times \frac{3(x - 2)}{5} = 10 \times \frac{2x + 1}{10}$$

$$2 \times 3(x - 2) = 2x + 1$$

Expand the bracket on the left:

$$2 \times (3x - 6) = 2x + 1$$

$$6x - 12 = 2x + 1$$

Collect like terms by subtracting $2x$ from both sides:

$$4x - 12 = 1$$

Add 12 to both sides:

$$4x = 13$$

Divide both sides by 4:

$$x = \frac{13}{4}$$

Check by substituting $x = \frac{13}{4}$ back into the original equation to verify.