Linear Equations · Grade 7
How do you solve a first degree equation?
An equation is a balance: as long as you do the same thing to both sides, the equality holds. Isolate the unknown using inverse operations — subtract what was added, divide by what multiplied — working in the reverse of the order of operations. Then substitute the solution back to check it.
Learning objectives
- Solve an equation of the form x plus a number equals a number, and of the form a number times x equals a number
- Isolate the variable with inverse operations taken in the right order
- Check a solution by substituting it back into the original equation
The balance
An equation says the two sides are equal, like two pans of a balance at rest. Take three kilos off one pan only and the balance is broken.
So every operation is done to both sides. Subtract the same number from both, divide both by the same number — and the equality stays true at every step.
This is not a trick; it is the only rule in the topic. Everything that looks like a further rule — moving a term across, for instance — is a consequence of it.
Inverse operations, in reverse order
Every operation is undone by its inverse: addition by subtraction, multiplication by division. To uncover the unknown, undo what was done to it.
The order is the reverse of the usual order of operations. In three x plus five, the x was multiplied and then added to — so subtract the five first and only then divide by three.
The check
Once you have a solution, substitute it into the original equation and confirm both sides come out equal. It is a ten-second check that catches every sign error.
Substitute into the original, not into the last line you wrote. If a mistake crept in partway, every line after it will confirm that mistake, and the original is the only one that will not let it through.
Worked examples
Solve: x plus 7 equals 12
- 7 was added to x, so the inverse is subtraction
- Subtract 7 from both sides
- The left leaves x, the right leaves 12 minus 7
Answer: x equals 5. Check: 5 plus 7 is 12
Solve: 3x equals 21
- x was multiplied by 3, so the inverse is division
- Divide both sides by 3
- The left leaves x, the right leaves 21 over 3
Answer: x equals 7. Check: 3 times 7 is 21
Solve: 4x minus 5 equals 11
- Undo the subtraction first: add 5 to both sides
- This gives 4x equals 16
- Divide both sides by 4
Answer: x equals 4. Check: 4 times 4 minus 5 is 11
Common mistakes
- Doing the operation to one side only
- That breaks the equality, and everything written afterwards is no longer about the same equation. The solution that comes out will not survive substitution.
- Dividing before subtracting
- The order is the reverse of the order of operations. In three x plus five equals twenty, dividing early means dividing the five as well — an extra step that tends to get forgotten.
- Checking against the last line instead of the original
- Each line follows from the one before, so a mistake partway is confirmed by everything after it. Only the original equation is a real check.
What to remember
- The same operation on both sides, always.
- Undo with the inverse operation.
- Work in the reverse of the order of operations.
- Substitute back into the original equation.