Solving One- and Two-Step Equations

An equation says that two things are equal. In x+7=19, the letter x stands for a number we don't know yet. To solve the equation, find the number that makes both sides equal. Here x=12, because 12+7=19.

A one-step equation needs one operation to find the letter, as in 5m=20. A two-step equation needs two, as in 3x+5=26.

In the exam you are usually asked to "Solve" a one-step equation for 1 mark, or a two-step equation for 2 marks, often in the first half of the paper. The answer goes on a line that starts with the letter, such as x=

The hardest ones on Foundation papers have a letter divided by a number, or an answer that is negative. Some questions are in words, and you write the equation yourself first.

Think of an equation as a balance. Whatever you do to one side, do exactly the same to the other side, and it stays balanced. Write the new equation on the next line each time.

An inverse operation undoes another one: - Subtracting undoes adding, and adding undoes subtracting. - Dividing undoes multiplying, and multiplying undoes dividing.

Remember that 5m means 5\times m, and \frac{x}{3} means x\div3.

In \frac{c}{5}=8, the letter has been divided by 5. Undo it by multiplying both sides by 5: c=8\times5=40

Check: 40\div5=8. Dividing 8 by 5 instead would undo the wrong operation.

In 3x+5=26, two things happen to x: it is multiplied by 3, then 5 is added. That makes 26.

To get back to x, undo the last operation first. Subtract 5 from both sides, then divide both sides by 3.

Write each step as a new equation on its own line. Writing 26-5\div3 in one go is wrong, because the division would be done first.

The answer does not have to be a whole number, or positive. 2x=7 gives x=3.5, and x+10=4 gives x=-6

When you take a bigger number from a smaller one, the answer is negative. Use a number line if it helps: 3-15=-12

If the x term is taken away, as in 20-3x=11, add 3x to both sides first: 20=11+3x. Then subtract 11: 9=3x, so x=3

Choose a letter for the amount you don't know, and say what it stands for. Write the total using the letter, and set it equal to the total you are given.

Then solve the equation, and give the answer to the question that was asked, with its units.

• Doing an operation to one side only. Do the same to both sides, and write the new equation.
• Using the wrong inverse. For 3y=15, divide by 3; don't multiply by 3 or subtract 3.
• Undoing in the wrong order. Undo the adding or subtracting first, then the multiplying or dividing.
• Dividing when the letter was divided. For \frac{x}{5}=2, multiply both sides by 5.
• Losing the minus sign. 2-8 is -6, not 6; keep the sign through to the answer.
• Guessing and checking. A wrong guess scores no marks, so use inverse operations and write each step.