Learn the balance method, then use it with division, negative coefficients and fractional solutions. Substitute your answer into the original equation to check.
- Keep an equation balanced by applying the same operation to both sides.
- Solve equations involving a constant and a multiplication or division, including negative values.
- Check an exact solution by substituting it into the original equation.
The idea
An equation states that two expressions have equal value. Applying the same reversible operation to both sides preserves that equality. For an equation with a multiple of the variable plus a constant on the left, undo the addition or subtraction first, then the multiplication.
Before you begin: Use inverse operations and substitute a number into an expression.
Worked examples
01 Start with the method
Solve the equation, then check your solution.
Subtract 5 from both sides to remove the constant term.
Divide both sides by 3. The unknown is now isolated.
Substitute 5 into the original equation. Both sides equal 20, so the solution works.
A mistake to avoid
A term does not simply jump to the other side. Name the operation you apply to both sides; this prevents sign errors when the constant is negative.
02Undoing division as well as subtraction
Solve the equation and verify the result.
Add three to both sides. This removes the subtraction outside the division.
Multiply both sides by four to undo division by four.
Check in the original equation, including the subtraction.
03When the variable has a negative coefficient
Solve and check. Keep track of the sign attached to the variable term.
Subtract seven from both sides. The variable term left behind is negative two times x.
Divide both sides by negative two, not positive two.
Substitute negative four into the original equation. Subtracting a negative product makes the total larger.
Practice questions with answers
Work these out first, then open each solution to compare your method. These questions are untimed.
Question 1
Solve and check this equation.
Show solution 1
Add 7 to both sides, divide by 4, then check: four lots of 4 minus 7 is 9.
Question 2
Solve for x.
Show solution 2
Subtract three to get twice x equals eight, then divide by two. Checking gives two times four plus three equals eleven.
Question 3
Solve and check the subtraction.
Show solution 3
Add eight to both sides to obtain five times x equals fifteen. Divide by five. In the original equation, fifteen minus eight equals seven.
Question 4
Solve an equation whose solution is negative.
Show solution 4
Subtract four from both sides, giving x divided by three equals negative two. Multiply by three. Negative six divided by three plus four is two.
Question 5
Give the exact solution as a fraction.
Show solution 5
Subtract nine to obtain negative four times x equals six. Dividing by negative four gives negative six fourths, or negative three halves. Substitution gives nine plus six equals fifteen.
Common questions
Do I always subtract first when solving a two-step equation?
No. Undo the constant operation using its inverse: subtract an added constant or add a subtracted constant. Then undo multiplication or division. The aim is to isolate the variable while keeping both sides equal, not to follow a fixed subtraction rule.
Can a negative or fractional answer be correct?
Yes. A linear equation can have either. Keep the exact value, then substitute it into the original equation. If both sides have the same value, the solution works; an unexpected sign is a reason to check, not a reason to discard it.