Linear Equations · Grade 7
How do you solve an equation with brackets and with fractions?
Expand the brackets first, then clear the denominators by multiplying the whole equation by the common denominator. The multiplication has to reach every term on both sides, not only the fractions — that is the mistake which ruins most exercises in this topic. After those two steps a familiar equation is left.
Learning objectives
- Solve an equation that contains brackets on one side or on both
- Clear denominators by multiplying both sides by the common denominator
- Solve an equation that contains brackets and fractions together
- Spot the common mistake of multiplying only one term of a side
Brackets first
While brackets remain you cannot collect and you cannot move terms. Expand them with the distributive law, on both sides, and only then carry on.
Watch especially for a minus before brackets — it flips the sign of every term inside, and this is where it gets dropped more than anywhere else.
Clearing denominators
Fractions in an equation do not force you to work in fractions. You can be rid of them in one move: multiply the whole equation by the common denominator of every denominator appearing in it.
The multiplication has to reach every term on both sides, including terms with no fraction in them at all. Multiplying only the fractions changes the equation, and the answer that comes out will not survive substitution.
After the multiplication each denominator cancels and a whole-number equation is left — exactly the kind you already know how to solve.
The order that works
Brackets, then denominators, then gather the unknown, then isolate. That order works on every exercise in the topic and requires no decision at each step.
At the end, substitute into the original equation, fractions and brackets and all. It is also the only check that catches a partial multiplication, because it is the one thing that did not pass through the mistake.
Worked examples
Solve: 3(x − 2) equals 12
- Expand: 3x minus 6
- Add 6 to both sides: 3x equals 18
- Divide by 3
Answer: x equals 6. Check: 3 times 4 is 12
Solve: x over 2 plus x over 3 equals 5
- The common denominator of 2 and 3 is 6
- Multiply every term by 6, including the 5
- This gives 3x plus 2x equals 30
- Collect and divide: 5x equals 30
Answer: x equals 6
Solve: (x + 1) over 4 equals 2 minus x over 2
- The common denominator is 4
- Multiply every term by 4: x plus 1 equals 8 minus 2x
- Gather: 3x equals 7
Answer: x equals 7 over 3
Common mistakes
- Multiplying only the terms that contain a fraction
- The multiplication is of the whole equation, so it applies to whole terms too. Leaving one out changes the equation, and substituting into the original exposes it.
- Clearing denominators before expanding brackets
- Brackets hide terms that also need multiplying. Expanding first prevents that entirely.
- Multiplying by the denominator of just one fraction
- The common denominator has to cover every denominator in the equation. Multiplying by some of them leaves a fraction behind and complicates what was meant to simplify.
What to remember
- Brackets first, denominators second.
- Multiply every term on both sides.
- Use the common denominator of them all.
- Check against the original equation.