Practice Maths

Simultaneous Equations

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Key Terms

Simultaneous equations
are two (or more) equations that must be satisfied by the same values of x and y at the same time.
The solution is the pair (x, y) that satisfies both equations — graphically, it is the point where the two lines intersect.
Substitution method
rearrange one equation to get y = … (or x = …), then substitute into the other equation.
Elimination method
add or subtract multiples of the equations to cancel one variable.
Special cases
parallel lines have no solution (inconsistent); the same line has infinitely many solutions (dependent).
📚 Two Methods at a Glance
Substitution: best when one equation is already solved for a variable (e.g. y = 2x + 1).
Elimination: best when both equations are in ax + by = c form, especially when coefficients match or are easily scaled.
Always check: substitute your (x, y) answer back into BOTH original equations.
x y 1 2 3 4 5 −1 1 2 3 4 −1 2x − y = 4 x + y = 5 Solution: (3, 2)

Worked Example 1 — Substitution method

Solve: y = 2x − 3   and   3x + y = 12

Step 1 — Equation (1) is already solved for y. Substitute into Equation (2):

   3x + (2x − 3) = 12

   5x − 3 = 12

   5x = 15  →  x = 3

Step 2 — Substitute x = 3 back into Equation (1):

   y = 2(3) − 3 = 6 − 3 = 3

Solution: x = 3, y = 3   i.e. (3, 3)

Check in Eq (2): 3(3) + 3 = 9 + 3 = 12 ✓

Worked Example 2 — Elimination method

Solve: 3x + 2y = 16   and   5x − 2y = 8

Step 1 — The y-coefficients are +2 and −2. Add the equations to eliminate y:

   (3x + 2y) + (5x − 2y) = 16 + 8

   8x = 24  →  x = 3

Step 2 — Substitute x = 3 into Equation (1):

   3(3) + 2y = 16  →  9 + 2y = 16  →  2y = 7  →  y = 3.5

Solution: x = 3, y = 3.5   i.e. (3, 3.5)

Check in Eq (2): 5(3) − 2(3.5) = 15 − 7 = 8 ✓

Hot Tip — For elimination, you may need to multiply one or both equations by a constant to make coefficients match. E.g., to solve 2x + y = 7 and 5x + 3y = 19: multiply the first by 3 → 6x + 3y = 21, then subtract the second → x = 2. Always check your solution in BOTH original equations, not just one.

Introduction

When two unknowns are related by two different conditions, we have a system of simultaneous equations. A single equation with two unknowns has infinitely many solutions (every point on the line). A second equation gives us a second condition — and the unique point satisfying both conditions is the intersection of the two lines.

The Substitution Method

Substitution works by reducing the system to a single equation with one unknown. The key idea: express one variable in terms of the other, then “substitute” (replace) it in the second equation.

When to use substitution:

Use substitution when one variable is already isolated (e.g. y = 3x − 1) or easily isolated (e.g. x + y = 5 can be rearranged to y = 5 − x).

System: y = 4 − x   and   2x + 3y = 18

Substitute y = 4 − x: 2x + 3(4 − x) = 18 → 2x + 12 − 3x = 18 → −x = 6 → x = −6

Back-substitute: y = 4 − (−6) = 10    Solution: (−6, 10)

The Elimination Method

Elimination works by adding or subtracting multiples of the equations to make one variable disappear. You may need to multiply one (or both) equations by a constant first.

Scaling to match coefficients:

System: 2x + 3y = 13   and   4x − y = 5

Multiply Eq (2) by 3: 12x − 3y = 15

Add to Eq (1): 2x + 3y + 12x − 3y = 13 + 15 → 14x = 28 → x = 2

Substitute into Eq (2): 4(2) − y = 5 → 8 − y = 5 → y = 3    Solution: (2, 3)

Special Cases

No solution (parallel lines): After elimination, you get a false statement like 0 = 5. This means the lines are parallel — they never intersect. Check by comparing gradients: if m&sub1; = m&sub2; but c&sub1; ≠ c&sub2;, there is no solution.

Infinite solutions (same line): After elimination, you get a true statement like 0 = 0. The equations describe the same line. Every point on the line is a solution.

Word Problems — Setting Up the System

Read the problem carefully and identify two unknown quantities (assign variables). Write two equations, one for each condition stated. Solve, then answer the question in context.

Example: Two numbers

Two numbers sum to 30 and their difference is 8. Find them.

Let the numbers be x and y: x + y = 30   and   x − y = 8

Add the equations: 2x = 38 → x = 19    Subtract: y = 30 − 19 = 11

The two numbers are 19 and 11.

💡 Choosing the method: Substitution is often faster when one variable is isolated. Elimination is more efficient when both equations are in standard form and coefficients are easy to match. For most exam questions, either method works — choose the one that feels cleaner for the given system.

Summary

Simultaneous equations: find (x, y) satisfying both equations simultaneously. Methods: substitution (isolate one variable, substitute) or elimination (add/subtract to cancel a variable). Graphically: the intersection point. Special cases: parallel lines (no solution) or identical lines (infinite solutions). Always check by substituting back into both original equations.

Mastery Practice

See Answers ➔
  1. Fluency

    Solve each system by substitution.

    1. (a) y = x + 2   and   3x + y = 10
    2. (b) y = 3 − 2x   and   x + 2y = 6
  2. Fluency

    Solve each system by substitution.

    1. (a) 2x − y = 5   and   y = x − 1
    2. (b) x = 4y + 1   and   2x − 3y = 8
  3. Fluency

    Solve each system by elimination.

    1. (a) 3x + 2y = 12   and   x + 2y = 8
    2. (b) 5x − 3y = 11   and   5x + y = 15
  4. Fluency

    Solve each system by elimination.

    1. (a) 2x + 3y = 13   and   4x − 3y = 11
    2. (b) 3x + 5y = 22   and   6x + 5y = 37
  5. Understanding

    For each system, choose the most efficient method (substitution or elimination), state your choice, and solve.

    1. (a) y = 2x   and   4x + 3y = 50
    2. (b) 2x + 3y = 18   and   3x + 2y = 17
  6. Understanding

    Determine the number of solutions for each system without fully solving. Explain your reasoning.

    1. (a) 2x + 4y = 10   and   x + 2y = 5
    2. (b) 3x − y = 7   and   3x − y = 2
  7. Understanding

    Word problem — two numbers.

    The sum of two numbers is 47 and their difference is 13. Write a system of equations and solve to find both numbers.

  8. Understanding

    Word problem — café sales.

    A café sells coffee at $4.50 and tea at $2.80. In one hour they sold 45 drinks for a total of $165.50. Write a system of equations and solve to find how many coffees and how many teas were sold.

  9. Problem Solving

    Two train stations are 180 km apart. Train A departs Station 1 travelling toward Station 2 at 60 km/h. Train B departs Station 2 travelling toward Station 1 at 80 km/h. Both leave at the same time.

    1. (a) Write equations for the position of each train at time t hours after departure (measuring distance from Station 1 in km).
    2. (b) Find when and where the trains meet by solving simultaneously.
    3. (c) How far from Station 1 do they meet?
  10. Problem Solving

    A company produces two products: X and Y.

    Product X: requires 3 hours labour + 2 hours machine time.
    Product Y: requires 1 hour labour + 4 hours machine time.
    Available per day: 24 hours labour, 32 hours machine time.
    1. (a) Define variables and write a system of equations assuming all available hours are used.
    2. (b) Solve the system to find how many of each product are made per day.
    3. (c) If Product X earns $80 profit and Product Y earns $120 profit, calculate the total daily profit.