Solving equations: one step and two step equations

An equation is a balance

An equation says two things are equal. Think of a balance scale with the same weight on each side. If you take 5 away from one side, the scale tips, unless you also take 5 away from the other. Solving an equation means doing the same thing to both sides until x is alone on one side.

In x + 5 = 12, the left pan holds a mystery bag, x, and 5 marbles. The right pan holds 12 marbles. Take 5 marbles off each pan and the bag balances 7 marbles, so x = 7.

One step equations

A one step equation needs a single inverse operation. Adding and subtracting undo each other, and so do multiplying and dividing. Remember that 6x means 6 × x, so dividing by 6 undoes it.

• x + 18 = 45: subtract 18 from both sides, so x = 27.
• x − 9 = 14: add 9 to both sides, so x = 23.
• 6x = 54: divide both sides by 6, so x = 9.
• x ÷ 4 = 7: multiply both sides by 4, so x = 28.
• Decimals and fractions use the same steps: y + 2.5 = 8.1 gives y = 5.6, and (2/3)x = 10 gives x = 15.

Two step equations

A two step equation has two operations to undo. Undo them in reverse order, like taking off shoes and socks: the last thing done is the first thing undone. In 3x − 4 = 17, x was multiplied by 3 and then 4 was subtracted, so add 4 first to get 3x = 21, then divide by 3 to get x = 7.

• Negative answers are fine: 2x + 15 = 9 gives 2x = −6, so x = −3.
• A negative coefficient: in 20 − 3x = 5, subtract 20 from both sides to get −3x = −15, then divide by −3, so x = 5.
• A division first: in x/5 + 2 = 6, subtract 2 to get x/5 = 4, then multiply by 5, so x = 20.

Always check your answer

Putting your answer back into the original equation is the fastest way to know you are right. For 2x + 15 = 9 with x = −3: 2 × (−3) + 15 = −6 + 15 = 9. Both sides match, so x = −3 is correct. If they don't match, the slip is in one of your two steps, and it is usually easy to find.

Solving equations word problems

In a word problem, choose a letter for the unknown and write the equation in the order the story happens. A phone plan costs $15 a month plus $3 per GB of data, and one month's bill is $36. If g is the number of GB, then 15 + 3g = 36. Subtract 15 to get 3g = 21, then divide by 3: g = 7 GB.

A rectangle 6 cm wide has a perimeter of 40 cm. With length L, 2L + 12 = 40, so 2L = 28 and L = 14 cm. Check: 14 + 6 + 14 + 6 = 40.

How it is taught in school

In the US, Common Core teaches that solving an equation means finding the value that makes it true (6.EE.B.5). Grade 6 students solve one step equations of the form x + p = q and px = q (6.EE.B.7). Grade 7 students solve word problems leading to equations of the form px + q = r and p(x + q) = r, where the numbers can be negative or fractions (7.EE.B.4.a).

In England, Year 6 pupils express missing number problems algebraically and find pairs of numbers that satisfy an equation with two unknowns. Solving linear equations with algebraic methods, including ones that need rearranging, is part of Key Stage 3.

How parents can help

• Play a think-of-a-number game: I think of a number, multiply it by 3 and add 2, and get 20. What was my number? That is 3x + 2 = 20, solved by working backward.
• Before your child starts, ask them to say what was done to x, in order. Undoing is easy once the order is clear.
• Insist on the check. Substituting the answer takes seconds and catches most errors.
• Encourage neat working: one step per line, with the same operation written on both sides.
• Keep sessions short. Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.