Solve linear equations in one unknown, including those with brackets and fractions, and form linear equations from word problems

What this dot point is asking

SEAB wants you to solve linear equations in one unknown (equations where the unknown appears only to the first power), including ones with brackets and fractions, and to form a linear equation from a word problem and solve it. Solving equations is the engine of algebra: graphs, geometry and statistics all lead back to equations.

The answer

The balance method

An equation says two expressions are equal. To keep it true, whatever you do to one side you must do to the other. The goal is to get the unknown by itself on one side.

Use inverse operations to undo what has been done to the unknown:

• The inverse of adding is subtracting, and the reverse.
• The inverse of multiplying is dividing, and the reverse.

For $3x + 5 = 20$, subtract $5$ from both sides to get $3x = 15$, then divide both sides by $3$ to get $x = 5$.

Unknowns on both sides

When the unknown appears on both sides, first collect the unknown terms on one side and the numbers on the other. For $7x - 4 = 3x + 12$, subtract $3x$ from both sides to get $4x - 4 = 12$, add $4$ to get $4x = 16$, then divide by $4$ to get $x = 4$.

Equations with brackets

Expand the brackets first, then solve as usual. For $2(x + 3) = 14$, expand to $2x + 6 = 14$, subtract $6$ to get $2x = 8$, then divide by $2$ to get $x = 4$.

Equations with fractions

Multiply every term by the denominator to clear the fraction. For $\dfrac{x}{3} = 4$, multiply both sides by $3$ to get $x = 12$. For $\dfrac{x + 1}{2} = 5$, multiply both sides by $2$ to get $x + 1 = 10$, so $x = 9$.

Checking your solution

Substitute your answer back into the original equation. If both sides give the same value, the solution is correct. This catches sign and arithmetic slips and is quick to do.

Forming equations from words

Translate a word problem in three steps: let a letter stand for the unknown, write the relationship as an equation, then solve it. Phrases such as "is", "gives" or "equals" mark where the equals sign goes.

Examples in context

Example 1. An age problem. Mei is $3$ years older than her brother, and together their ages add to $21$. Let the brother be $x$, so Mei is $x + 3$ and $x + (x + 3) = 21$. This gives $2x + 3 = 21$, so $2x = 18$ and $x = 9$. The brother is $9$ and Mei is $12$. Defining one unknown and writing the other in terms of it keeps the equation linear.

Example 2. A geometry link. Two angles on a straight line are $2x$ and $x + 30$. Since angles on a straight line sum to $180^\circ$, we have $2x + x + 30 = 180$, so $3x = 150$ and $x = 50$. Many geometry questions reduce to a linear equation once the angle fact is written down.

Try this

• Cue. Solve $4x + 9 = 33$. Subtract $9$ to get $4x = 24$, then divide by $4$ to get $x = 6$.
• Cue. Solve $3(x - 2) = 18$. Expand to $3x - 6 = 18$, add $6$, divide by $3$ to get $x = 8$.
• Cue. Solve $\dfrac{x}{5} + 1 = 4$. Subtract $1$ to get $\dfrac{x}{5} = 3$, then multiply by $5$ to get $x = 15$.