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Simultaneous equations0%

Algebra · Topic 9 of 12

Simultaneous equations

Video lesson5 worked examples

Theory

You must be able to construct simultaneous equations from a written context and solve them algebraically or graphically.

Contexts often include real-life situations such as comparing costs (e.g., hiring a car, mobile phone charges) or finding the exact intersection of two lines.

They can also be solved graphically: draw both lines on the same grid (a quick way is to find where each crosses the axes), and the coordinates of the point where they cross are the solution.

The Golden Rule: to eliminate a variable by adding or subtracting, its coefficients must match. Scale one or both equations until they do — then add if the matching signs are opposite, subtract if they are the same. Always substitute back to find the second variable.

⚠️ Common Examiner Traps

Add vs subtract: if the matching terms have the same sign, subtract; if opposite signs, add. Getting this wrong is the usual error.
Only scaling one equation: sometimes both equations must be multiplied (by different numbers) to make a pair of coefficients match.
Forgetting the second variable: finding one value is half the answer — substitute back to get the other.
Sign slips when subtracting: subtracting a negative becomes an addition, e.g. $7 - (-2) = 9$ .

Worked examples

Example 1

Elimination (coefficients already match)

Solve the system $3x + 2y = 12$ and $5x - 2y = 4$ .

Step 1: The $y$ terms are $+2y$ and $-2y$ — opposite signs, so add the equations to eliminate $y$ :

(3x + 5x) + (2y - 2y) = 12 + 4 \implies 8x = 16

Step 2: So $x = 2$ . Substitute back into the first equation:

3(2) + 2y = 12 \implies 2y = 6 \implies y = 3

Answer: $x = 2,\ y = 3$ .

Example 2

Elimination (scaling both equations)

Solve the system $2x + 3y = 13$ and $3x + 2y = 12$ .

Step 1: No coefficients match, so scale both to make the $x$ terms equal. Multiply the first by 3 and the second by 2:

6x + 9y = 39 \qquad 6x + 4y = 24

Step 2: The $x$ terms now match with the same sign, so subtract:

(9y - 4y) = 39 - 24 \implies 5y = 15 \implies y = 3

Step 3: Substitute back into $2x + 3y = 13$ : $2x + 9 = 13 \implies x = 2$ .

Answer: $x = 2,\ y = 3$ .

Example 3

🎯 Exam-style (constructing from a context)

"4 apples and 2 bananas cost £2.40. 3 apples and 4 bananas cost £2.30." Find the cost of one apple (a) and one banana (b).

Step 1: Construct equations: $4a + 2b = 240$ and $3a + 4b = 230$ (working in pence).

Step 2: Scale equations to eliminate b. Multiply the first by 2: $8a + 4b = 480$ .

Step 3: Subtract the second equation from the scaled first equation: $(8a - 3a) = 480 - 230 \rightarrow 5a = 250 \rightarrow a = 50$ .

Step 4: Substitute $a = 50$ back into an original equation: $4(50) + 2b = 240 \rightarrow 200 + 2b = 240 \rightarrow 2b = 40 \rightarrow b = 20$ .

Answer: One apple costs £0.50, one banana costs £0.20.

Example 4

Solving Graphically

Solve the system $x + y = 6$ and $y = 2x$ graphically.

Step 1: Find two points on each line so you can draw them. For $x + y = 6$ , the axis crossings are quickest: $(0, 6)$ and $(6, 0)$ .

Step 2: For $y = 2x$ , take $(0, 0)$ and $(2, 4)$ .

Step 3: Draw both lines on the same grid and read off where they cross — the point of intersection is $(2, 4)$ .

Step 4: Check in both equations: $2 + 4 = 6$ ✓ and $4 = 2(2)$ ✓.

Answer: $x = 2,\ y = 4$ .

Example 5

🎯 Exam-style (point of intersection)

Two straight lines have equations $2x + y = 11$ and $3x - y = 9$ . Find, algebraically, the coordinates of their point of intersection.

Step 1: The point of intersection is the pair of values satisfying both equations — so solve them simultaneously. The $y$ terms are $+y$ and $-y$ , so add the equations:

(2x + 3x) + (y - y) = 11 + 9 \implies 5x = 20 \implies x = 4

Step 2: Substitute back into the first equation:

2(4) + y = 11 \implies y = 3

Step 3: The question asks for coordinates, so give the answer as a point, not as two separate values.

Answer: the lines intersect at $(4,\ 3)$ .

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13 Simultaneous Equations questions with answers and video solutions