Practice Maths

Simplifying and Solving Equations

Key Ideas

Key Terms

Multi-step simplification
Simplifying an expression by first expanding all brackets, then collecting like terms.
Collecting like terms
Combining terms with the same variable and power into a single term (e.g. 5x + 3x = 8x).
Expanding brackets
Applying the distributive law to remove brackets: a(b + c) = ab + ac.
Equations with fractions
Equations containing fractional terms; solved by multiplying every term by the LCM of all denominators.
LCM
Lowest Common Multiple — the smallest number that is a multiple of two or more given numbers; used to clear fractions.
Geometric equations
Equations formed from angle or length relationships in geometry (e.g. angles on a straight line sum to 180°).

Multi-step Simplification

Expand brackets using the distributive law, then collect like terms: 3(2x + 4) − 2(x − 1) = 6x + 12 − 2x + 2 = 4x + 14.

Equations with Fractions

Find the LCM of all denominators. Multiply every term by this LCM to eliminate fractions before solving.

Hot Tip For equations like x/3 + x/4 = 7, the LCM of 3 and 4 is 12. Multiply all terms by 12: 4x + 3x = 84, so 7x = 84 and x = 12. Always check by substituting back.

Worked Example

Question: Solve (x + 2)/4 = (x − 1)/3.

LCM of 4 and 3 is 12. Multiply both sides by 12:
3(x + 2) = 4(x − 1)
3x + 6 = 4x − 4
6 + 4 = 4x − 3x
x = 10   Check: (10+2)/4 = 3, (10−1)/3 = 3 ✓

Collecting Like Terms

Before solving an equation, you often need to simplify each side first. Like terms are terms that have exactly the same variable (or no variable at all). You can add or subtract like terms, but you cannot combine unlike terms.

Examples of like terms: 3x and 5x (both have x); 7 and −2 (both are constants). Examples of unlike terms: 3x and 5x2 (different powers); 4x and 4y (different variables).

Simplify: 5x + 3 − 2x + 7 = (5x − 2x) + (3 + 7) = 3x + 10.

Expanding Brackets Before Solving

If an equation contains brackets, your first step is usually to expand them using the distributive law: a(b + c) = ab + ac.

Example: Solve 3(x + 4) = 21.
Step 1: Expand. 3x + 12 = 21.
Step 2: Subtract 12 from both sides. 3x = 9.
Step 3: Divide both sides by 3. x = 3.
Check: 3(3 + 4) = 3 × 7 = 21. ✓

When expanding with a negative outside the bracket, remember to change the sign of every term inside: −2(x − 5) = −2x + 10.

Solving Equations with Variables on Both Sides

When the variable appears on both sides of the equation, move all the variable terms to one side (usually the side where the variable ends up positive) and all the constants to the other side.

Example: Solve 5x − 3 = 2x + 9.
Step 1: Subtract 2x from both sides: 3x − 3 = 9.
Step 2: Add 3 to both sides: 3x = 12.
Step 3: Divide by 3: x = 4.
Check: 5(4) − 3 = 17; 2(4) + 9 = 17. ✓

Always move the smaller variable term to avoid negatives if you can. If you end up with a negative coefficient (like −2x = 6), just divide by the negative number: x = −3.

Checking Solutions by Substitution

The most reliable way to verify your answer is to substitute it back into the original equation and check that both sides are equal. This is a quick, two-second check that can save you marks in an exam.

If after substitution the left side equals the right side, your answer is correct. If they don't match, go back and find the error — common mistakes include sign errors when expanding brackets or moving terms.

Example: After solving 4(2x − 1) = 3x + 11, you get x = 3. Check: LHS = 4(2 × 3 − 1) = 4 × 5 = 20. RHS = 3 × 3 + 11 = 9 + 11 = 20. LHS = RHS. ✓

Multi-Step Equations: Putting It Together

For complex equations, follow this order: (1) expand all brackets, (2) collect like terms on each side, (3) move variable terms to one side, (4) move constants to the other side, (5) divide both sides to isolate the variable, (6) check by substitution.

Example: Solve 2(3x + 1) − 4 = 5x + 8.
Expand: 6x + 2 − 4 = 5x + 8.
Simplify LHS: 6x − 2 = 5x + 8.
Subtract 5x: x − 2 = 8.
Add 2: x = 10.
Check: LHS = 2(31) − 4 = 62 − 4 = 58; RHS = 50 + 8 = 58. ✓

Key tip: The golden rule of equations is: whatever you do to one side, you must do exactly the same to the other side. This keeps the balance. A common mistake is expanding −2(x − 3) as −2x − 6 (wrong!) instead of −2x + 6 (correct). Always be careful with signs when expanding brackets that have a negative out the front.

Mastery Practice

  1. Expand and collect like terms to simplify each expression. Fluency

    1. 3(x + 4) + 2x
    2. 5(2x − 1) − 3x
    3. 4(a + 3) + 2(a − 1)
    4. 6(b − 2) − 3(b + 1)
    5. 2(3m + 4) + 5(m − 2)
    6. 4(2p − 3) − 2(p − 5)
    7. 3(x + 2) + 4(2x − 1) − 5x
    8. 7 − 2(3k − 4) + k

  2. Solve each equation. Show all working. Fluency

    1. 3x + 7 = 22
    2. 5x − 4 = 16
    3. 2(x + 3) = 14
    4. 3(2x − 1) = 15
    5. 4x + 5 = 2x + 13
    6. 7x − 3 = 4x + 9
    7. 5(x − 2) = 3x + 4
    8. 3(2x + 1) = 2(x + 9)

  3. Solve each equation. Multiply through by the LCM to clear fractions first. Fluency

    1. x/3 + 5 = 9
    2. x/4 − 2 = 3
    3. (x + 2)/4 = 3
    4. (2x − 1)/3 = 5
    5. x/2 + x/3 = 10
    6. x/4 + x/6 = 5
    7. x/3 + x/4 = 7
    8. x/5 − x/10 = 3

  4. Set up and solve an equation for each geometric situation. Understanding

    1. A rectangle has perimeter 38 cm. Its length is (2x + 1) cm and its width is (x − 2) cm. Find x and the dimensions.
    2. Two angles on a straight line are (3x + 10)° and (2x − 5)°. Find x.
    3. Three angles of a triangle are x°, (2x + 15)° and (x − 3)°. Find x and each angle.
    4. A square has perimeter (8x − 4) cm and a side length of (2x + 3) cm. Find x.
    5. Angles at a point sum to 360°. Three angles are (4x)°, (5x − 12)°, and (3x + 24)°. Find x.
    6. An isosceles triangle has two equal sides of (3x − 1) cm and a third side of (x + 7) cm. If the perimeter is 31 cm, find x.
    7. Two co-interior angles are (3x + 15)° and (2x + 5)°. They sum to 180°. Find x and each angle.
    8. A triangle has exterior angle (4x + 10)°. The two non-adjacent interior angles are (2x − 5)° and (x + 25)°. Find x.

  5. Write an equation for each problem and solve it. Understanding

    1. I think of a number, multiply it by 4 and subtract 7. The result is 21. What is the number?
    2. Five more than twice a number equals three less than five times the number. Find the number.
    3. The sum of three consecutive integers is 57. Find the integers.
    4. The sum of three consecutive even integers is 78. Find the integers.
    5. One number is 8 more than another. Their sum is 46. Find both numbers.
    6. One number is three times another. Their difference is 16. Find both numbers.
    7. Half of a number added to one third of the same number gives 20. Find the number.
    8. When a number is subtracted from twice itself, the result is one quarter of 48. Find the number.

  6. Apply your equation-solving skills to these real-world problems. Problem Solving

    1. A plumber charges $80 call-out fee plus $45 per hour. A painter charges $50 call-out fee plus $55 per hour.
      1. Write an equation for each tradesperson’s total cost C in terms of hours h.
      2. After how many hours do they charge the same amount?
      3. Who is cheaper for a 4-hour job?
    2. A rectangle’s length is 5 cm more than twice its width. The perimeter is 88 cm.
      1. Let the width be w cm. Write an equation for the perimeter.
      2. Solve to find the dimensions.
      3. Calculate the area.
    3. Jordan has twice as many marbles as Sam. After Jordan gives Sam 12 marbles, they have the same number.
      1. Let Sam have x marbles. Write expressions for their totals after the transfer.
      2. Solve to find how many marbles each person started with.
    4. A school fundraiser sold adult tickets for $12 and student tickets for $7. A total of 80 tickets were sold and $770 was raised.
      1. Let a = number of adult tickets. Write two equations.
      2. Use substitution to find the number of each type of ticket sold.

  7. Solve each problem by forming and solving an equation. Problem Solving

    Geometry challenge. Each situation involves unknown dimensions that must be found using an equation.
    1. A triangle has angles in the ratio 1 : 2 : 3. Find each angle. What type of triangle is it?
    2. Two supplementary angles differ by 28°. Find both angles.
    3. A rectangular paddock has perimeter 96 m. Its length is 6 m more than three times its width. Find the dimensions and area.
    4. The sum of the interior angles of a polygon is 900°. Six of the angles are each (2x + 10)° and two are each (x − 5)°. Find x and all the angles.

  8. Use equations to answer each question about the table. Problem Solving

    Mobile data costs. Two plans charge differently per month.
    GB usedPlan A cost ($)Plan B cost ($)
    0100
    21820
    42640
    63460
    1. Write an equation for each plan’s cost C in terms of GB used g.
    2. Solve to find at what usage the plans cost the same.
    3. A customer uses 8 GB per month. How much does each plan cost? Which is better?
    4. For what usage is Plan A cheaper than Plan B? Write and solve an inequality.

  9. Solve these challenging equation problems. Show all working. Problem Solving

    1. The sum of four consecutive odd integers is 104. Find the integers.
    2. A fraction has numerator (x + 1) and denominator (2x − 1). When simplified, the fraction equals ⅔. Find x and the original fraction.
    3. Two trains start from the same station and travel in opposite directions. Train A travels at (3x + 10) km/h and Train B at (5x − 4) km/h. After 2 hours, they are 220 km apart. Find x and the speed of each train.
    4. Anna is 3 times as old as Ben. In 8 years, Anna will be twice as old as Ben. Write two equations and solve to find their current ages.

  10. Apply multi-step equation solving to these extended problems. Problem Solving

    Extended modelling. Each problem requires forming and solving equations from written descriptions.
    1. A school canteen sells pies for $3.50 and sandwiches for $2.00. On Monday, 120 items were sold for a total of $315. Let p = number of pies sold.
      1. Write two equations using p and s (number of sandwiches).
      2. Solve the system to find the number of each item sold.
      3. On Tuesday the same totals apply but the prices swap. Which item now gives more revenue?
    2. A water tank is being filled and drained at the same time. A tap fills it at 12 L per minute and a drain removes water at (2x − 4) L per minute. The net fill rate is 6 L per minute.
      1. Write and solve an equation to find x.
      2. If the tank holds 900 L, how long to fill it from empty at the net rate?
    3. A rectangle has perimeter P = 2(l + w). If l = 3w − 2 and the area is 65 cm², set up and solve a quadratic (or trial-and-error) equation to find integer dimensions that satisfy both the area condition and l = 3w − 2. (Hint: try whole number values of w.)
    4. A savings challenge: Mia saves $x per week. After 6 weeks she has $240. She then increases her weekly saving by $15. How many more weeks until she has $600 total?
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