Practice Maths

Problem Solving with Simultaneous Equations

Key Ideas

Key Terms

Define your unknowns clearly
State what each variable represents, including units.
Translate
The problem into two equations by identifying two separate relationships between the unknowns.
Verify
Your answer satisfies both equations and makes sense in the real-world context (e.g. no negative numbers of people).

A Problem-Solving Framework

1. Read carefully — identify the unknowns.
2. Define variables with units.
3. Write two equations from the given information.
4. Solve using substitution or elimination.
5. Check in both equations and in the original context.
6. Answer the question in a sentence.

Hot Tip Read the question twice. The first read gives you the big picture; the second read helps you pick out the two relationships you need to write your equations.

Setting Up Equations from Word Problems

The hardest part of word problems is translating the English into algebra. Use this structured approach every time: (1) Define your variables — write down clearly what each letter represents, including units. (2) Write two equations — look for two separate facts or relationships stated in the problem. (3) Solve using substitution or elimination. (4) Interpret — state your answer in words using the context of the problem. (5) Check — verify the solution makes sense and satisfies both original conditions.

Cost and Ticket Problems

A classic example: "Adult tickets cost $12 and child tickets cost $8. A total of 50 tickets were sold for $520. How many of each type were sold?" Let a = number of adult tickets, c = number of child tickets. Equation 1 (total tickets): a + c = 50. Equation 2 (total revenue): 12a + 8c = 520. Solve: from equation 1, a = 50 − c. Substitute: 12(50 − c) + 8c = 520 → 600 − 12c + 8c = 520 → −4c = −80 → c = 20. So a = 30. Check: 30 + 20 = 50 ✓ and 12(30) + 8(20) = 360 + 160 = 520 ✓.

Age Problems and Mixture Problems

Age problems involve current ages and ages at a different time. Key phrase: "in 5 years" means add 5 to the current variable. For example, "Sam is twice as old as Alex. In 4 years, Sam will be 1.5 times Alex's age. Find their ages." Let s and a be current ages. Equations: s = 2a and (s + 4) = 1.5(a + 4). Substitute: 2a + 4 = 1.5a + 6, so 0.5a = 2, a = 4. Sam is 8.

Mixture problems involve combining two quantities (e.g. solutions of different concentrations, or items at different prices) to reach a target total. Write one equation for the total quantity and one for the total value or concentration.

Key tip: Before writing any equations, read the problem twice and underline every number and every relationship. Each underlined relationship should become part of an equation. If you cannot find two separate relationships, re-read the problem — it is always there. Most errors in word problems come from writing only one equation and guessing the second, or confusing which variable is which.

Break-Even Problems

Break-even analysis is a real-world application of simultaneous equations used in business. A business has a cost equation (fixed costs plus variable costs: C = fixed + rate × units) and a revenue equation (R = selling price × units). The break-even point is where C = R, meaning neither profit nor loss is made. Set the two equations equal and solve for the number of units. This is the intersection point of the two linear graphs.

Mastery Practice

  1. Mixture and blend problems. Define variables, write two equations, solve and verify. Understanding

    1. A juice bar makes a tropical blend using mango juice and pineapple juice. A 2-litre blend uses 400 mL more mango than pineapple. How many millilitres of each juice are in the blend?
    2. A nut mix contains cashews and almonds. The total mass is 800 g. There are three times as many almonds as cashews by mass. How many grams of each nut are in the mix?
    3. A chemist mixes a 20% acid solution with a 50% acid solution to make 300 mL of a 30% acid solution. Let x = volume of 20% solution and y = volume of 50% solution. Write two equations and solve.

  2. Cost, price and value problems. Understanding

    1. A market stall sells small bags of popcorn for $3 and large bags for $5. In one hour they sell 40 bags for a total of $148. How many of each size were sold?
    2. A mobile phone plan charges a monthly fee plus a cost per gigabyte of data. In January, Priya used 3 GB and paid $35. In February, she used 7 GB and paid $55. Find the monthly fee and the cost per gigabyte.
    3. At a theme park, an adult entry is $a and a child entry is $c. Two adults and three children paid $87. One adult and five children paid $85. Find the adult and child entry prices.

  3. Distance, speed and time problems. Use distance = speed × time. Problem Solving

    1. Two cyclists start from the same point and travel in opposite directions. After 3 hours they are 138 km apart. One cyclist travels 4 km/h faster than the other. Find each cyclist’s speed.
    2. A boat travels 60 km downstream in 3 hours and 60 km upstream in 5 hours. Let b be the boat’s speed in still water (km/h) and c be the current speed (km/h). Write two equations and find both speeds.
    3. A train and a car travel between two towns 240 km apart. The train takes 2 hours and the car takes 3 hours. How much faster is the train than the car?

  4. Age problems. Problem Solving

    1. Maya is 3 times as old as her brother Luca. In 8 years, Maya will be twice Luca’s age. How old are Maya and Luca now?
    2. The sum of a parent’s and child’s ages is 46. Five years ago, the parent was 5 times the child’s age. Find their current ages.
    3. Two brothers have ages that add to 25. Four years ago the older brother was double the younger brother’s age. How old is each brother now?

  5. Geometry problems. Problem Solving

    1. Two angles in a triangle are supplementary (they add to 180°) with one angle 30° more than the other. Find both angles. (Note: this is one pair of angles in the triangle.)
    2. The perimeter of a rectangle is 54 cm. The length is 9 cm more than the width. Find the dimensions.
    3. In a quadrilateral, two angles are unknown. The first angle is four times the second angle. Together with the known angles of 80° and 60°, all four angles sum to 360°. Find the two unknown angles.
    4. The angles of a triangle are x°, y° and 70°. The largest angle is 20° more than twice the smallest. Find x and y (angle sum of triangle = 180°).

  6. Work and rate problems. Define variables and write two equations. Problem Solving

    1. Alex and Sam together can paint a fence in 3 hours. Alex alone takes 5 hours. How long would Sam take alone? (Hint: use rates — fraction of the job completed per hour.)
    2. A printing press and a photocopier together produce 480 pages per hour. The printing press produces 3 times as many pages per hour as the photocopier. How many pages per hour does each machine produce?
    3. A factory has two assembly lines. Line A and Line B together produce 200 units per day. If Line A produced 20% more and Line B produced 10% less, the total would still be 200 units per day. How many units per day does each line currently produce?

  7. Coin and ticket counting problems. Problem Solving

    1. A jar contains 50-cent coins and $1 coins. There are 24 coins in total and their total value is $17. How many of each coin are there?
    2. Admission to a science expo costs $12 for adults and $7 for students. A school group of 35 people paid $300 in total. How many adults and how many students were in the group?

  8. A coffee shop blends two types of coffee beans. Premium beans cost $28 per kg and standard beans cost $14 per kg. The blend is to be 5 kg and cost $18 per kg on average. Problem Solving

    Mixture setup. Write one equation for the total mass and one equation for the total cost. Solve, then verify the average cost per kg.
    1. Let p = kg of premium beans and s = kg of standard beans. Write the mass equation.
    2. Write the cost equation.
    3. Solve to find how many kilograms of each type are needed.

  9. A mobile game charges a sign-up fee plus a monthly subscription. Jordan signed up and paid for 3 months, spending $47 in total. Her friend Riley joined later with a different plan: no sign-up fee but a monthly rate $3 more than Jordan’s. Riley paid $45 for 3 months. Find each sign-up fee and monthly rate. Problem Solving

    1. Define variables for Jordan’s sign-up fee and monthly rate. Write Jordan’s cost equation.
    2. Write Riley’s cost equation in terms of the same variables.
    3. Solve the system and state each person’s sign-up fee and monthly rate.

  10. Two cars start from towns A and B, which are 300 km apart, and travel toward each other at the same time. Car 1 travels at a constant speed. Car 2 travels 20 km/h faster than Car 1. They meet after 2.5 hours. Problem Solving

    Extension. After finding both speeds, determine how far each car has travelled when they meet, and how far each car is from its starting town.
    1. Let v km/h be Car 1’s speed. Write two equations using the meeting condition.
    2. Solve for both speeds.
    3. Find how far from Town A the cars meet.
    4. A third car starts from Town A at the same time as the others, but travels in the same direction as Car 2 at 90 km/h. After how many hours does Car 2 lap this third car? (Hint: when do Car 2 and the third car have the same displacement from Town B?)
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