Simultaneous Equations Worksheet — GCSE — Intermediate
Simultaneous Equations worksheet for Key Stage 4 / GCSE (Ages 14-16). Intermediate level maths practice, aligned to the UK National Curriculum. Print-ready with answer key included.
Mastering Simultaneous Equations
Key Stage 4 / GCSE (Ages 14-16) - Intermediate Level
Instructions: This worksheet will help you practise solving simultaneous equations using various methods. Each activity is designed to enhance your understanding and problem-solving skills. Work through each problem carefully and ensure all steps are shown clearly.
Solving Simultaneous Equations by Substitution
Use the substitution method to solve the following pairs of simultaneous equations. Substitute one equation into the other to find the values of x and y.
1. y = 2x + 3
3x + 2y = 12
2. x = 4y - 5
2x + 3y = 19
Solving Simultaneous Equations by Elimination
Apply the elimination method to solve these equations. Eliminate one variable by adding or subtracting the equations, and then solve for the remaining variable.
1. 2x + 3y = 13
4x - 3y = 7
2. 5x + 4y = 32
3x - 4y = 4
Word Problems Involving Simultaneous Equations
Translate the following word problems into simultaneous equations and solve them. Clearly define your variables and show all working.
1. The sum of two numbers is 10, and their difference is 2. Find the numbers.
2. A rectangle's length is twice its width. If the perimeter is 36 cm, find the dimensions of the rectangle.
Graphical Method for Solving Simultaneous Equations
Plot the following pairs of equations on graph paper and identify the point of intersection, which represents the solution to the equations.
1. y = x + 2
y = -x + 4
2. y = 2x - 1
y = -0.5x + 3
Checking Solutions of Simultaneous Equations
For each pair of equations, a potential solution is given. Verify if the solution satisfies both equations.
1. x = 2, y = 3 for 3x + 4y = 18 and 2x - y = 1
2. x = 1, y = 5 for x + 2y = 11 and 3x - y = -2
Creating Your Own Simultaneous Equations
Create a pair of simultaneous equations that have the solution x = 3 and y = -2. Write your equations and verify the solution.
Equation 1:
Equation 2:
Verification:
Simultaneous Equations with Fractions
Solve the following simultaneous equations that include fractions. Carefully handle the fractions to find the correct solution.
1. (1/2)x + (1/3)y = 5
(1/4)x - (1/5)y = 1
2. (2/3)x + (3/4)y = 7
(1/6)x + (1/2)y = 3
Simultaneous Equations in Real-Life Contexts
Read the following scenarios and form simultaneous equations to solve the problems. Show all your work.
1. A shop sells pens for £1.50 each and pencils for £0.50 each. If a customer buys 10 items for £10, how many of each did they buy?
2. Two cars start from the same point. Car A travels east at 60 km/h and Car B travels north at 80 km/h. After 2 hours, how far apart are the cars?
Activity 1: x = 1, y = 5 ; x = 3, y = 2
Activity 2: x = 5, y = 1 ; x = 4, y = 3
Activity 3: 6 and 4 ; Length = 12 cm, Width = 6 cm
Activity 4: (1,3) ; (1.6,2.2)
Activity 5: True ; False
Activity 6: Example: 2x + y = 4 ; x - 2y = 7 ; Verification: True
Activity 7: x = 15, y = 6 ; x = 9, y = 6
Activity 8: 5 pens and 5 pencils ; 200 km
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