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Year 10 Maths

Simultaneous Equations

Solve pairs of linear equations using graphical, substitution, and elimination methods.

What Are Simultaneous Equations?

Simultaneous equations are two or more equations that share the same unknowns. The solution is the set of values that satisfy all equations at the same time.

Example System

y = 2x + 1

y = −x + 7

We need to find the values of x and y that make both equations true.

Method 1: Graphical

Plot both lines on the same graph. The point of intersection is the solution.

0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 (2, 5) y = 2x + 1 y = −x + 7

The two lines intersect at (2, 5), so x = 2 and y = 5 is the solution.

Method 2: Substitution

Use one equation to express one variable in terms of the other, then substitute it into the second equation.

Equations: y = 2x + 1   (1)      y = −x + 7   (2)

Step 1: Since both equal y, set them equal: 2x + 1 = −x + 7

Step 2: Solve for x: 3x = 6, so x = 2

Step 3: Substitute x = 2 into equation (1): y = 2(2) + 1 = 5

Solution: x = 2, y = 5

Method 3: Elimination

Add or subtract the equations to eliminate one variable. You may need to multiply one or both equations first.

Equations: 2x + 3y = 12   (1)      4x − 3y = 6   (2)

Step 1: Notice the 3y terms are opposite signs. Add the equations:

(2x + 3y) + (4x − 3y) = 12 + 6

6x = 18, so x = 3

Step 2: Substitute x = 3 into equation (1): 2(3) + 3y = 12 ⇒ 6 + 3y = 12 ⇒ 3y = 6 ⇒ y = 2

Solution: x = 3, y = 2

Special Cases

No Solution (Parallel Lines)

If the lines are parallel (same gradient, different y-intercept), they never intersect.

Example: y = 2x + 1 and y = 2x + 5. Same gradient (2) but different intercepts — no solution.

Infinite Solutions (Same Line)

If both equations describe the same line, every point on the line is a solution.

Example: y = 3x + 2 and 2y = 6x + 4 (the second is just double the first).

Key Vocabulary

Term Definition
Simultaneous equations Two or more equations with shared unknowns solved at the same time.
Substitution Replacing a variable in one equation using an expression from another.
Elimination Adding or subtracting equations to remove one variable.
Point of intersection The point where two lines cross on a graph; the graphical solution.
Consistent / Inconsistent Consistent = at least one solution; inconsistent = no solution (parallel lines).

Worked Examples

1

Substitution: y = 3x − 1 and 2x + y = 9

Step 1: Substitute y = 3x − 1 into 2x + y = 9:

2x + (3x − 1) = 9

Step 2: Simplify: 5x − 1 = 9 ⇒ 5x = 10 ⇒ x = 2

Step 3: Find y: y = 3(2) − 1 = 5

Solution: x = 2, y = 5

2

Elimination: 3x + 2y = 16 and x + 2y = 10

Step 1: Subtract equation (2) from equation (1) to eliminate y:

(3x + 2y) − (x + 2y) = 16 − 10

2x = 6 ⇒ x = 3

Step 2: Substitute x = 3 into equation (2): 3 + 2y = 10 ⇒ 2y = 7 ⇒ y = 3.5

Solution: x = 3, y = 3.5

3

Elimination with multiplication: 2x + 5y = 21 and 3x + 2y = 12

Step 1: Multiply (1) by 3 and (2) by 2 to match x-coefficients:

6x + 15y = 63   (1′)

6x + 4y = 24   (2′)

Step 2: Subtract (2′) from (1′): 11y = 39 ⇒ y = 39/11 ≈ 3.55

Step 3: Substitute back: 3x + 2(39/11) = 12 ⇒ 3x = 12 − 78/11 = 54/11 ⇒ x = 18/11 ≈ 1.64

Solution: x = 18/11, y = 39/11 (or approximately x ≈ 1.64, y ≈ 3.55)

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see feedback.

Question 1

Solve: y = x + 3 and y = 2x + 1. What is the solution?

Question 2

Solve by elimination: x + y = 8 and x − y = 2. What is x?

Question 3

The equations y = 4x − 1 and y = 4x + 3 have:

Question 4

Solve: 3x + y = 10 and x = 4. What is y?

Question 5

Which method is best when one equation is already solved for y (e.g. y = 2x + 5)?

Key Concepts Summary