AC9M7A03 · Year 7 · Algebra

Solving one-variable linear equations

ACARA v9 CONTENT DESCRIPTION “solve one-variable linear equations with natural number solutions; verify the solution by substitution”

An equation is a statement that two things are equal, and solving one means finding the value of the unknown that makes the statement true. Equations are how mathematics answers questions like what number, when doubled and increased by 3, gives 11. This year you learn to solve simple linear equations and to check your answers, a skill that sits at the very centre of algebra.

A linear equation has a single variable that is not raised to any power, such as 2x plus 3 equals 11. Solving it means working out what x must be. The most reliable way to think about the process is to picture a balance scale, because that image keeps every step honest.

An equation is a balance

Imagine a scale with 2x plus 3 in the left pan and 11 in the right. Because the equation says these are equal, the scale sits perfectly level. This picture gives you the golden rule of solving equations: whatever you do to one side, you must do to the other. Add to one pan only, or take from one pan only, and the balance is lost. Treat both sides identically and the scale stays level, which means the equation stays true.

An equation balances
The two sides of an equation weigh the same, like a level scale.
An equation says two things are equal, like a balanced scale. With 2x plus 3 on one side and 11 on the other, the scale is level. Whatever you do to one side you must do to the other to keep it balanced.

Keeping the balance is what makes solving trustworthy. Every legal move, adding the same amount to both sides, subtracting the same, multiplying or dividing both by the same number, preserves the equality. This is why you can transform a complicated-looking equation into a simple one without ever changing which value of x makes it true.

Solving by undoing operations

To find x, you peel away the operations surrounding it, using inverse operations, until x stands alone. In 2x plus 3 equals 11, the x has been multiplied by 2 and then had 3 added. Undo these in reverse: first subtract 3 from both sides to get 2x equals 8, then divide both sides by 2 to get x equals 4. Each step keeps the scale balanced, and step by step the variable is freed.

Solving by undoing
Reverse the operations one at a time to leave the variable by itself.
To solve an equation, undo the operations around the variable, doing the same to both sides. Take 3 off each side, then divide each side by 2, leaving x equals 4. Substituting 4 back in confirms it, since 2 times 4 plus 3 really is 11.

The final step, and the one most often skipped, is to check the answer by substituting it back into the original equation. Putting x equals 4 into 2x plus 3 gives 2 times 4 plus 3, which is 11, exactly the right-hand side. The check confirms the solution and catches any slip made along the way. Solving equations by keeping a balance and undoing operations, then verifying by substitution, is a method that will carry you through all of the algebra still to come.

Teaching tip: a real or drawn balance scale makes the golden rule unforgettable. Place objects representing each side and show that removing weight from only one pan tips it over. Linking the physical tipping to the idea of an equation breaking gives the rule a reason rather than leaving it as something to memorise.

Encourage the habit of always checking by substitution, even when the answer feels obvious. It turns equation solving into a self-correcting process, and a student who routinely verifies will catch their own errors long before anyone else needs to point them out.

Builds on: Formulating algebraic expressions (AC9M7A02). That unit built expressions; this unit sets an expression equal to a value and solves for the variable.
Quick self-check
1. What does the equals sign in an equation tell you?
2. To solve x + 5 = 12, what do you do to both sides?
3. Solve 3x = 18.
4. What is the value of x in 2x + 1 = 9?
5. Why is it useful to substitute your answer back into the original equation?