Simultaneous Equations
Solving systems of linear and quadratic equations using elimination and substitution
SPEC 2.4 • ALGEBRA AND FUNCTIONS
What Are Simultaneous Equations?
Definition
Simultaneous equations are two or more equations that share the same variables and must be solved together to find values that satisfy ALL equations at the same time.
In this lesson, we'll solve:
Two linear equations — using elimination or substitution
One linear + one quadratic — using substitution only
Key Idea
The solution(s) are the values of x and y that make both equations true simultaneously.
Two Methods for Linear Equations
When solving two linear equations, we have two methods to choose from:
Method 1
ELIMINATION
Add or subtract equations to eliminate one variable
Method 2
SUBSTITUTION
Rearrange one equation and substitute into the other
Which Method?
Elimination: Best when coefficients are easy to match by multiplying
Substitution: Best when one equation is already solved for a variable (e.g., y = 2x + 1)
Elimination Method
Method
Label the equations (1) and (2)
Multiply one or both equations so the coefficients of one variable match
Add or subtract the equations to eliminate that variable
Solve for the remaining variable
Substitute back to find the other variable
Check your answer in both original equations
Important
When subtracting equations, be careful with signs! A common error is forgetting to change signs when subtracting the second equation.
Worked Example — Elimination01
Solve the simultaneous equations:
1
Label equations: (1) 2x + 3y = 12 and (2) x − y = 1
2
Multiply (2) by 2: 2x − 2y = 2 ... call this (3)
3
Subtract (3) from (1): (2x + 3y) − (2x − 2y) = 12 − 2
4
Simplify: 5y = 10, so y = 2
5
Substitute into (2): x − 2 = 1, so x = 3
6
Check in (1): 2(3) + 3(2) = 6 + 6 = 12 ✓
Answer: x = 3, y = 2
Worked Example — Elimination02
Solve the simultaneous equations:
1
Label equations: (1) 3x + 4y = 18 and (2) 5x + 3y = 19
2
To eliminate y, multiply (1) by 3 and (2) by 4:
•
(1) × 3: 9x + 12y = 54
•
(2) × 4: 20x + 12y = 76
3
Subtract: (20x + 12y) − (9x + 12y) = 76 − 54
4
Simplify: 11x = 22, so x = 2
5
Substitute into (1): 3(2) + 4y = 18, so 6 + 4y = 18
6
Solve: 4y = 12, so y = 3
7
Check in (2): 5(2) + 3(3) = 10 + 9 = 19 ✓
Answer: x = 2, y = 3
Substitution Method
Method
Rearrange one equation to make x or y the subject
Substitute this expression into the other equation
Solve the resulting equation
Substitute back to find the other variable
Check your answer in both original equations
Tip
This method is particularly efficient when one equation is already in the form y = ... or x = ... — no rearranging needed!
Worked Example — Substitution03
Solve the simultaneous equations:
1
Equation (2) already has y as subject: y = 2x − 3
2
Substitute into (1): 3x + 2(2x − 3) = 7
3
Expand: 3x + 4x − 6 = 7
4
Simplify: 7x = 13, so x = 13/7
5
Substitute back: y = 2(13/7) − 3 = 26/7 − 21/7 = 5/7
Answer: x = 13/7, y = 5/7
Worked Example — Substitution04
Solve the simultaneous equations:
1
Rearrange (2) to make y the subject: y = 2x
2
Substitute into (1): x + 3(2x) = 7
3
Simplify: x + 6x = 7, so 7x = 7
4
Solve: x = 1
5
Substitute back: y = 2(1) = 2
6
Check in (1): 1 + 3(2) = 1 + 6 = 7 ✓
Answer: x = 1, y = 2
Linear and Quadratic Equations
When one equation is linear and one is quadratic (e.g., a parabola), we must use substitution.
Method
Rearrange the linear equation to make x or y the subject
Substitute into the quadratic equation
Expand and simplify to get a quadratic equation
Solve the quadratic (factorise or use the formula)
Find the corresponding values of the other variable
Write solutions as coordinate pairs
Important
The quadratic may have 0, 1, or 2 solutions — make sure to find ALL solutions and pair them correctly!
Worked Example — Line and Parabola05
Find the points of intersection of:
and
1
Set equal (both = y): x + 2 = x² − 3x + 3
2
Rearrange: 0 = x² − 4x + 1
3
Quadratic formula: x = (4 ± √(16 − 4))/2 = (4 ± √12)/2 = 2 ± √3
4
Find y-values using y = x + 2:
•
When x = 2 + √3: y = 4 + √3
•
When x = 2 − √3: y = 4 − √3
Answer: (2 + √3, 4 + √3) and (2 − √3, 4 − √3)
Worked Example — Line and Parabola06
Find the points of intersection of:
and
1
Set equal: 2x + 1 = x² − x + 3
2
Rearrange: 0 = x² − 3x + 2
3
Factorise: (x − 1)(x − 2) = 0
4
So x = 1 or x = 2
5
Find y using y = 2x + 1:
•
When x = 1: y = 2(1) + 1 = 3
•
When x = 2: y = 2(2) + 1 = 5
Answer: (1, 3) and (2, 5)
Practice
Question 1: Solve the simultaneous equations using elimination:
and
Add the equations: (4x + 3y) + (2x − 3y) = 17 + 1
Simplify: 6x = 18, so x = 3
Substitute into equation 1: 4(3) + 3y = 17, so 12 + 3y = 17
Solve: 3y = 5, so y = 5/3
Answer: x = 3, y = 5/3
Practice
Question 2: Solve the simultaneous equations using substitution:
and
Substitute x = y + 1 into equation 1: 2(y + 1) + y = 8
Expand: 2y + 2 + y = 8
Simplify: 3y = 6, so y = 2
Substitute back: x = 2 + 1 = 3
Answer: x = 3, y = 2
Practice
Question 3: Find the points of intersection of:
and
Set equal: 3x − 1 = x² + x − 3
Rearrange: 0 = x² − 2x − 2
Quadratic formula: x = (2 ± √(4 + 8))/2 = (2 ± √12)/2 = 1 ± √3
Find y using y = 3x − 1:
When x = 1 + √3: y = 3(1 + √3) − 1 = 2 + 3√3
When x = 1 − √3: y = 3(1 − √3) − 1 = 2 − 3√3
Answer: (1 + √3, 2 + 3√3) and (1 − √3, 2 − 3√3)
Summary
Key Methods
TypeMethodWhen to Use
Two linearEliminationCoefficients easy to match
Two linearSubstitutionOne equation already solved for x or y
Linear + QuadraticSubstitutionAlways use substitution
Key Exam Skills
Choose the most efficient method for linear equations
Always use substitution for linear + quadratic
Remember to find BOTH variables
Write answers as coordinate pairs for intersection problems
Always check your solutions in both original equations
Next: What Are Simultaneous Equations?