ACARA v9 CONTENT DESCRIPTION “find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations”
Builds on: Rules That Grow Patterns (AC9M6A01). Using a letter for a position leads naturally to using one for an unknown — here the letter stands for a missing number to be found.
An equation is a balance
An equation states that two things are equal: whatever sits on the left of the equals sign has the same value as whatever sits on the right. Like a balance scale, it stays level only when both sides weigh the same. When one side contains an unknown, written as a letter or a box, solving the equation means finding the value that keeps the balance level. The equation x plus 3 equals 7 is asking a clear question: what number, added to 3, gives 7? Reading an equation as a balance is the idea the whole unit rests on.
An equation balances
An equation is a balance: whatever is on the left must equal what is on the right.
x + 3 = 7 — an equation says the two sides balance. What value of x keeps it level?
Inverse operations undo
To find an unknown, you undo whatever was done to it. Each operation has an inverse, an opposite that reverses it: addition and subtraction undo each other, and so do multiplication and division. If a number has had 6 added to it to make 15, subtracting 6 gets back to the number. The golden rule is to do the same thing to both sides, so the balance stays level. Applying the inverse to both sides is the single most important move in solving equations, and every later step is built from it.
Inverse operations undo
To find an unknown, undo what was done to it using the opposite operation.
x + 6 = 15. To free x, apply the inverse: − 6 on both sides.
Unknowns inside brackets
Some equations wrap the unknown inside brackets, like 3 times the quantity x plus 2 equals 21. Brackets are undone from the outside in. Here the whole bracket has been multiplied by 3, so dividing both sides by 3 leaves x plus 2 equals 7; then subtracting 2 gives x equals 5. Working outward first mirrors how the expression was built: the bracket was formed, then multiplied, so the multiplication is undone before the addition inside. Handling brackets this way lets equations with several steps be unwound one operation at a time.
Unknowns inside brackets
With brackets, undo the operation outside the brackets first, then work inward.
3 × (x + 2) = 21. Undo the outer × first, then the inner +.
Order of operations
To find an unknown correctly, you must read an expression the way everyone agrees to read it. The order of operations sets the rules: brackets first, then multiplication and division, then addition and subtraction. Without it, 2 plus 3 times 4 could be read two ways, but the agreed order gives 3 times 4 equals 12, then 2 plus 12 equals 14. The same rules decide how an equation with an unknown is unwound. Knowing the order of operations is what makes a numerical equation have one definite value rather than several possible ones.
Order of operations
Brackets first, then multiply and divide, then add and subtract.
2 + 3 × 4 = ? Multiply and divide before you add and subtract; brackets come first of all.
Finding the missing number
A number sentence with a box, like 5 times the box equals 35, is an equation in disguise. Finding the missing number means asking what value fits, and the inverse operations give the answer: 35 divided by 5 is 7, so the box is 7. These sentences come in every form, using each of the four operations and sometimes brackets, but the method is always the same. Identify what has been done to the unknown, then undo it. Solving for a missing number is the plain-language version of solving an equation.
Find the missing number
A box stands for the unknown. Find the value that makes the sentence true.
Find the value that makes 5 × □ = 35 true. Pick A, B or C.
Checking a solution
Every solution can be checked, and checking is a habit worth keeping. Once a value for the unknown is found, putting it back into the original equation shows whether both sides come out equal. If solving 3 times the quantity x plus 2 equals 21 gives x equals 5, then substituting 5 back gives 3 times 7, which is 21, matching the right-hand side exactly. When the two sides agree, the solution is confirmed; when they do not, it signals a slip to find. Substituting back turns a guess into a certainty.
Check by substituting back
Test a solution by putting it back into the equation and confirming both sides match.
A solution can be checked: put x = 5 back in and see if both sides come out equal.
From sentences to equations
Finding unknowns in number sentences is the doorway to algebra proper. The box that stands for a missing number becomes a letter standing for a variable, and the inverse operations used here become the methods for solving the equations of later years. Balancing both sides, respecting the order of operations, working through brackets, and checking by substitution are the exact skills that carry forward. What begins as filling in a missing number grows into the central activity of algebra: solving for the unknown.
Quick self-check
1. In the equation x + 6 = 15, what is the value of x?
2. Which operation undoes multiplying by 4?
3. Solve 3 × (x + 2) = 21 for x.
4. What does 2 + 3 × 4 equal?
5. You solve an equation and get x = 5. How can you check it is correct?