Equations - Guess and Check
Every question with a fully worked solution.
- Equations - Guess and Check - Video - Guess and Check Method Watch
Theory
An equation is a number sentence with an equals sign. Its solution is the number that makes both sides equal. This Year 7 maths page shows how to test a solution and how to solve equations by guess, check and improve, using each result to make a better guess.
\(3x + 5\) is an expression: it has no equals sign. \(3x + 5 = 26\) is an equation: it says the two sides are equal. The left-hand side (LHS) is \(3x + 5\) and the right-hand side (RHS) is \(26\).
Solving an equation means finding the value of the pronumeral that makes LHS \(=\) RHS. That value is the solution.
To test whether a number is a solution, substitute it. If LHS \(=\) RHS it is a solution; if not, it is not.
Guess, check, improve:
- Guess a value for the pronumeral.
- Substitute it and work out the LHS.
- Compare with the RHS: too small or too big? Use this to choose a better guess.
- Repeat until LHS \(=\) RHS, then check.
How to solve by guess and check
- Make a first guess and work out the LHS (order of operations).
- Decide: is the LHS too small or too big?
- Improve the guess in the right direction.
- Stop when LHS \(=\) RHS, then write the solution.
(ii) Is \(y = 4\) a solution of \(3y + 2 = 14\)?
| \(4 \times 6 - 7\) | \(=\) | \(17\) |
| \(3 \times 4 + 2\) | \(=\) | \(14\) |
(i) No: \(17 \ne 19\).
(ii) Yes: LHS \(= 14 =\) RHS.
| \(2 \times 4 + 4\) | \(=\) | \(12 \text{ (too small)}\) |
| \(2 \times 5 + 4\) | \(=\) | \(14 \;\checkmark\) |
\(x = 5\)
| \(5 \times 5 + 3\) | \(=\) | \(28 \text{ (too small)}\) |
| \(5 \times 8 + 3\) | \(=\) | \(43 \text{ (too big)}\) |
| \(5 \times 7 + 3\) | \(=\) | \(38 \;\checkmark\) |
The solution is between \(5\) and \(8\): \(x = 7\).
| \(14 - 2 \times 2\) | \(=\) | \(10 \text{ (too big)}\) |
| \(14 - 2 \times 3\) | \(=\) | \(8 \;\checkmark\) |
A bigger \(m\) makes the LHS smaller, so after \(10\) (too big) try a bigger \(m\). \(m = 3\).
(i) \(3(x - 2) = 21\)
(ii) \(\dfrac{a}{4} + 1 = 6\)
| \(3 \times (5 - 2)\) | \(=\) | \(9 \text{ (too small)}\) |
| \(3 \times (10 - 2)\) | \(=\) | \(24 \text{ (too big)}\) |
| \(3 \times (9 - 2)\) | \(=\) | \(21 \;\checkmark\) |
| \(\dfrac{20}{4} + 1\) | \(=\) | \(6 \;\checkmark\) |
(i) \(x = 9\)
(ii) Try multiples of \(4\): \(a = 20\).
The next two integers are \(x + 1\) and \(x + 2\), so \(x + (x + 1) + (x + 2) = 45\).
| \(10 + 11 + 12\) | \(=\) | \(33 \text{ (too small)}\) |
| \(15 + 16 + 17\) | \(=\) | \(48 \text{ (too big)}\) |
| \(14 + 15 + 16\) | \(=\) | \(45 \;\checkmark\) |
The integers are \(14\), \(15\) and \(16\).
Common pitfalls
Frequently asked questions
What is the difference between an expression and an equation?
An equation has an equals sign; an expression does not.
What is the solution of an equation?
The value of the pronumeral that makes the left-hand side equal to the right-hand side.
How do you check if a number is a solution?
Substitute it. If LHS \(=\) RHS, it is a solution.
How does guess and check work?
Guess, work out the left-hand side, see if it is too big or too small, then improve the guess.
What are consecutive integers?
Integers that follow each other, such as \(14\), \(15\), \(16\): \(x\), \(x + 1\), \(x + 2\).