An equation stands an unknown number as x and asks for which value of x the two sides of the equals sign are the same.
If sweets shared equally among three people give each person 4, the starting number of sweets x satisfies x ÷ 3 = 4, so it is 12.
Adding or subtracting the same number on both sides, or multiplying or dividing both sides by the same non-zero number, keeps the equation true, so you can use this to leave x on its own and find the solution.
On this page you put different numbers into x to see when the scale balances, then follow the four steps of solving by moving terms across.
An equation is a balance
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Balance scale: compare the two sides for the number put into x; the solution makes it level
2
3
11
2
👀 Properties of equality
①Adding the same number to both sides keeps an equation true
②Subtracting the same number from both sides keeps it true
③Multiplying both sides by the same number keeps it true
④Dividing both sides by the same non-zero number keeps it true
⑤On a scale, doing the same to both pans never upsets the balance. In the starting picture, 2x + 3 = 11 gives 7 < 11 for x = 2, so the right side goes down, and it levels out at x = 4
Solving step by step
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Four steps to solve the linear equation ax + b = c
🔍 The order of steps
①Move the constant term to the right-hand side; its sign changes as it crosses
②Work out the right side to get ax = (a number)
③Divide both sides by the coefficient a
④When you reach x = (a number), that number is the solution. If it is not a whole number, leave it as a fraction in its lowest terms
Moving terms across, and checking
Moving a term across
Move a term to the other side of the equation and change its sign
ax + b = c → ax = c − b
Checking
Put the solution back into the original equation and see whether both sides match
For 2x + 3 = 11 with x = 4, the left side is 2 × 4 + 3 = 11 and the right side is 11
💡 Moving terms is a shortcut for the properties of equality
①Subtracting b from both sides of ax + b = c gives ax = c − b
②Looking only at the result, the +b on the left seems to have crossed over and become −b on the right. That is what moving a term across means
③So it is not a new rule, just a short way of writing "subtract the same number from both sides"
④On the scale, taking b away from both pans leaves it balanced
Check with examples
Example 1
Solve the linear equation 2x + 3 = 11.
1
Move the constant 3 to the right-hand side.
2x = 11 − 3 = 8
2
Divide both sides by 2.
x = 8 ÷ 2 = 4
▸ x = 4
Check: 2 × 4 + 3 = 11, correct. On the scale, with a = 2, b = 3, c = 11, putting 4 into x makes it level.
Example 2
Solve 3(x − 2) = x + 4.
1
Expand the brackets, then move the x terms to the left and the constants to the right.
3x − 6 = x + 4 → 3x − x = 4 + 6
2
Simplify and divide by the coefficient of x.
2x = 10 → x = 5
▸ x = 5
Check: the left side is 3 × (5 − 2) = 9 and the right side is 5 + 4 = 9.
Exam checklist
Solving a linear equation
ax + b = c → x = c - ba
move terms → simplify → divide by the coefficient → check
Grade 7 school test style
The linear equation 2x + a = 5x − 7 has solution x = 3. What is the constant a?
①1
②2
③3
④4
⑤5
▸ ② 2
1
Putting the solution x = 3 into the equation must make it true.
2 × 3 + a = 5 × 3 − 7
2
Simplify and find a.
6 + a = 8 ⇒ a = 2
🎯 Exam checklist
①An equation needs an equals sign. Something like 2x + 3 with no equals sign is only an expression
②An equation is linear if, after moving every term to the left and simplifying, it takes the form (an expression of degree 1 in x) = 0
③When moving terms, make sure you changed the sign
④If coefficients include fractions or decimals, multiply both sides by a suitable number to make them whole first
⑤Get into the habit of checking your solution in the original equation