simultaneous equations problem solving worksheet

Solving Simultaneous Equations: Worksheets with Answers

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Simultaneous Equations

Here is everything you need to know about simultaneous equations for GCSE maths (Edexcel, AQA and OCR).

You’ll learn what simultaneous equations are and how to solve them algebraically. We will also discuss their relationship to graphs and how they can be solved graphically.

Look out for the simultaneous equations worksheets and exam questions at the end.

What are simultaneous equations?

Simultaneous equations are two or more algebraic equations that share variables such as x and y .

They are called simultaneous equations because the equations are solved at the same time.

The number of variables in simultaneous equations must match the number of equations for it to be solved.

An example of simultaneous equations is 2 x + 4 y = 14 4 x − 4 y = 4

Here are some more:

Each of these equations on their own could have infinite possible solutions.

However when we have at least as many equations as variables we may be able to solve them using methods for solving simultaneous equations.

Representing simultaneous equations graphically

We can consider each equation as a function which, when displayed graphically, may intersect at a specific point. This point of intersection gives the solution to the simultaneous equations.

When we draw the graphs of these two equations, we can see that they intersect at (1,5).

So the solutions to the simultaneous equations in this instance are:

x = 1 and y = 5

Solving simultaneous equations

When solving simultaneous equations you will need different methods depending on what sort of simultaneous equations you are dealing with. There are two sorts of simultaneous equations you will need to solve:

• linear simultaneous equations
• quadratic simultaneous equations

A linear equation contains terms that are raised to a power that is no higher than one.

Linear simultaneous equations are usually solved by what’s called the elimination method (although the substitution method is also an option for you ) .

Solving simultaneous equations using the elimination method requires you to first eliminate one of the variables, next find the value of one variable, then find the value of the remaining variable via substitution. Examples of this method are given below.

A quadratic equation contains terms that are raised to a power that is no higher than two.

Quadratic simultaneous equations are solved by the substitution method.

See also: 15 Simultaneous equations questions

What are linear and quadratic simultaneous equations?

Simultaneous equations worksheets

Get your free simultaneous equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

How to solve simultaneous equations

To solve pairs of simultaneous equations you need to:

• Use the elimination method to get rid of one of the variables.
• Find the value of one variable.
• Find the value of the remaining variables using substitution.
• Clearly state the final answer.
• Check your answer by substituting both values into either of the original equations.

How do you solve pairs of simultaneous equations?

See the examples below for how to solve the simultaneous linear equations using the three most common forms of simultaneous equations.

See also: Substitution

Quadratic simultaneous equations

Quadratic simultaneous equations have two or more equations that share variables that are raised to powers up to 2 e.g.  x^{2} and y^{2} . Solving quadratic simultaneous equations algebraically by substitution is covered, with examples, in a separate lesson.

Step-by-step guide: Quadratic simultaneous equations

Simultaneous equations examples

For each of the simultaneous equations examples below we have included a graphical representation.

Step-by-step guide : Solving simultaneous equations graphically

Example 1: Solving simultaneous equations by elimination (addition)

• Eliminate one of the variables.

By adding the two equations together we can eliminate the variable y .

2 Find the value of one variable.

3 Find the value of the remaining variable via substitution.

We know x = 3 so we can substitute this value into either of our original equations.

4 Clearly state the final answer.

5 Check your answer by substituting both values into either of the original equations.

This is correct so we can be confident our answer is correct.

Graphical representation of solving by elimination (addition)

When we draw the graphs of these linear equations they produce two straight lines. These two lines intersect at (1,5). So the solution to the simultaneous equations is x = 3 and y = 2 .

Example 2: Solving simultaneous equations by elimination (subtraction)

By subtracting the two equations we can eliminate the variable b .

NOTE: b − b = 0 so b is eliminated

3 Find the value of the remaining variable/s via substitution.

We know a = 2 so we can substitute this value into either of our original equations.

Graphical representation of solving by elimination (subtraction)

When graphed these two equations intersect at (1,5). So the solution to the simultaneous equations is a = 2 and b = 6 .

Example 3: Solving simultaneous equations by elimination (different coefficients)

Notice that adding or subtracting the equations does not eliminate either variable (see below).

This is because neither of the coefficients of h or i are the same. If you look at the first two examples this was the case.

So our first step in eliminating one of the variables is to make either coefficients of h or i the same.

We are going to equate the variable of h .

Multiply every term in the first equation by 2 .

Multiply every term in the second equation by 3 .

Now the coefficients of h are the same in each of these new equations, we can proceed with our steps from the first two examples. In this example, we are going to subtract the equations.

Note: 6h − 6h = 0 so h is eliminated

Careful : 16 − − 6 = 22

We know i = − 2 so we can substitute this value into either of our original equations.

Graphical representation of solving by elimination (different coefficients)

When graphed these two equations intersect at (1,5). So the solution to the simultaneous equations is h = 4 and i = − 2 .

Example 4: Worded simultaneous equation

David buys 10 apples and 6 bananas in a shop. They cost £5 in total. In the same shop, Ellie buys 3 apples and 1 banana. She spends £1.30 in total. Find the cost of one apple and one banana.

Additional step: conversion

We need to convert this worded example into mathematical language. We can do this by representing apples with a and bananas with b .

Notice we now have equations where we do not have equal coefficients (see example 3).

We are going to equate the variable of b .

Multiply every term in the first equation by 1 .

Multiply every term in the second equation by 6 .

Now the coefficients of b are the same in each equation we can proceed with our steps from the previous examples. In this example, we are going to subtract the equations.

NOTE: 6b − 6b = 0 so b is eliminated

16 − − 6 = 22

Note : we ÷ (− 8) not 8

We know a = 0.35 so we can substitute this value into either of our original equations.

1 apple costs £0.35 (or 35p ) and 1 banana costs £0.25 (or 25p ).

Graphical representation of the worded simultaneous equatio

When graphed these two equations intersect at (1,5). So the solution to the simultaneous equations is a = 0.35 and b = 0.25 .

Common misconceptions

• Incorrectly eliminating a variable. Using addition to eliminate one variable when you should subtract (and vice-versa).
• Errors with negative numbers. Making small mistakes when +, −, ✕, ÷ with negative numbers can lead to an incorrect answer. Working out the calculation separately can help to minimise error. Step by step guide: Negative numbers (coming soon)
• Not multiplying every term in the equation. Mistakes when multiplying an equation. For example, forgetting to multiply every term by the same number.
• Not checking the answer using substitution. Errors can quickly be spotted by substituting your solutions in the original first or second equations to check they work.

Practice simultaneous equations questions

1. Solve the Simultaneous Equation

6x +3y = 48 6x +y =26

Subtracting the second equation from the first equation leads to a single variable equation. Use this equation to determine the value of y , then substitute this value into either equation to determine the value of x .

2. Solve the Simultaneous Equation x -2y = 8 x -3y =3

Subtracting the second equation from the first equation leads to a single variable equation, which determines the value of y . Substitute this value into either equation to determine the value of x .

3. Solve the Simultaneous Equation 4x +2y = 34 3x +y =21

In this case, a good strategy is to multiply the second equation by 2 . We can then subtract the first equation from the second to leave an equation with a single variable. Once this value is determined, we can substitute it into either equation to find the value of the other variable.

4. Solve the Simultaneous Equation:

15x -4y = 82 5x -9y =12

In this case, a good strategy is to multiply the second equation by 3 . We can then subtract the second equation from the first to leave an equation with a single variable. Once this value is determined, we can substitute it into either equation to find the value of the other variable.

Simultaneous equations GCSE questions

1. Solve the simultaneous equations

\begin{array}{l} 5x=-10 \\ x=-2 \end{array}      or correct attempt to find y

One unknown substituted back into either equation

2. Solve the simultaneous equations

Correct attempt to multiple either equation to equate coefficients e.g.

Correct attempt to find y or x ( 16y=56 or 16x = 24 seen)

3. Solve the simultaneous equations

Correct attempt to find y or x ( 13x=91 or 13y=-39 seen)

Learning checklist

• Solve two simultaneous equations with two variables (linear/linear) algebraically
• Derive two simultaneous equations, solve the equation(s) and interpret the solution

The next lessons are

• Maths formulas
• Types of graphs
• Interpreting graphs

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Systems of Equations Worksheets | Simultaneous Equations

Packed in this compilation of printable worksheets on systems of equations are adequate exercises for 8th grade and high school students to check if the ordered pair is a solution to the pair of equations, determine the number of solutions, classify systems of equations as consistent, inconsistent, dependent or independent. Also, learn to solve a set of simultaneous equations with 2 and 3 variables using various methods. Begin solving systems of equations with our free worksheets!

Check whether the Ordered Pair is a Solution

Verify if the ordered pair is a solution to the system of equations by plugging it in the equations. Write 'yes' if the substitution proves both the equations true or else write 'no'. Make an attempt to write the systems of equations using the ordered pairs as well. This pdf resource is ideal for grade 8.

• Download the set

Unique solution, Infinite solutions or No Solution

State whether the systems of equations result in a unique solution, no solution or infinite solutions in this set of printable high school worksheets. Solve each pair of equations and label it based on the number of solutions.

Classify Consistent or Inconsistent

Figure out graphically or by using the coefficients, if each system of equations has a solution or not. Label the equation as 'consistent' if it has at least one solution and 'inconsistent' if it has no solution.

Determine Dependent or Independent

Consistent equations are classified as independent when they have exactly one solution with two lines intersecting at a point graphically; and dependent when they have infinite solutions and two superimposed lines when represented graphically.

Solving the Systems of Equations with 2 Variables Worksheets

Employ methods like graphing, substitution, cross-multiplication, elimination, Cramer's Rule to solve pairs of simultaneous equations with 2 variables. Find PDFs to solve reciprocal equations as well.

(24 Worksheets)

Solving Systems of Equations with 3 Variables Worksheets

Bolster skills with this batch of high school worksheets on solving simultaneous equations featuring 3 variables.

(15 Worksheets)

Related Worksheets

» Rearranging Equations

» Linear Equation of a Line

» One-step Equation

» Two-Step Equation

» Multi-Step Equation

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Unit 1: Linear simultaneous equations

Geometrical representation.

• Solutions to systems of equations: dependent vs. independent (Opens a modal)
• Number of solutions to a system of equations (Opens a modal)

Solution of simultaneous equations by use of graphs

• Systems of equations with graphing: 5x+3y=7 & 3x-2y=8 (Opens a modal)
• Number of solutions to a system of equations graphically Get 3 of 4 questions to level up!

Conditions of solvability of two linear simultaneous equations

• No videos or articles available in this lesson
• Number of solutions of system of equations Get 3 of 4 questions to level up!
• Number of solutions to systems of equations (intermediate) Get 3 of 4 questions to level up!

Method of substitution

• Systems of equations with substitution: -3x-4y=-2 & y=2x-5 (Opens a modal)
• Systems of equations with substitution Get 3 of 4 questions to level up!

Method of elimination

• Systems of equations with elimination (and manipulation) (Opens a modal)
• Systems of equations with elimination Get 3 of 4 questions to level up!
• Systems of equations with elimination challenge Get 3 of 4 questions to level up!

Cross multiplication

• Solving system of equations through cross multiplication Get 3 of 4 questions to level up!

Equations reducible to linear form

• Identifying substitutions: Equations reducible to linear form Get 3 of 4 questions to level up!
• Solving equations reducible to linear form Get 3 of 4 questions to level up!
• Word problems: Writing equations reducible to linear form Get 3 of 4 questions to level up!

Linear equations word problems

• Age word problem: Ben & William (Opens a modal)
• Forming equations with two variables Get 3 of 4 questions to level up!
• Age word problems Get 3 of 4 questions to level up!
• Word problems involving pair of linear equations (advanced) Get 3 of 4 questions to level up!
• Systems of equations word problems Get 3 of 4 questions to level up!

Worksheet on Simultaneous Linear Equations

Practice each pair of the equation problems from the worksheet on simultaneous linear equations with the two variables and two linear equations. Solving simultaneous linear equations with two variables by using the method of substitution to solve each pair of the equations and also solve the equations by using the method of elimination.

1.  Use the method of substitution to solve each other of the pair of simultaneous equations:    (a) x + y = 15                     x - y = 3

  (b) x + y = 0                       x - y = 2

  (c) 2x - y = 3                    4x + y = 3

  (d) 2x - 9y = 9               5x + 2y = 27

  (e) x + 4y = -4               3y - 5x = -1

  (f) 2x - 3y = 2                  x + 2y = 8

  (g) x + y = 7                  2x - 3y = 9

  (h) 11y + 15x = -23        7y - 2x = 20

   (i) 5x - 6y = 2                6x - 5y = 9

2. Solve each other pair of equation given below using elimination method:   (a) x + 2y = -4                        3x - 5y = -1

  (b) 4x + 9y = 5                      -5x + 3y = 8

  (c) 9x - 6y = 12                      4x + 6y = 14

  (d) 2y - (3/x) = 12                   5y + (7/x) = 1

  (e) (3/x) + (2/y) = (9/xy)       (9/x) + (4/y) = (21/xy)

  (f) (4/y) + (3/x) = 8             (6/y) + (5/x) = 13

  (g) 5x + (4/y) = 7                    4x + (3/y) = 5

  (h) x + y = 3                         -3x + 2y = 1

  (i) -3x + 2y = 5                       4x + 5y = 2

3. Solve the following simultaneous equations:   (a) 3a + 4b = 43                      -2a + 3b = 11

  (b) 4x - 3y = 23                      3x + 4y = 11

  (c) 5x + (4/y) = 7                    4x + (3/y) = 5

  (d) 4/(p - 3) + 6/(q - 4) = 5       5/(p - 3) - 3/(q - 4) = 1

  (e) (l/6) - (m/15) = 4               (l/3) - (m/12) = 19/4

  (f) 3x + 2y = 8                        4x + y = 9

  (g) x - y = -1                         2y + 3x = 12

  (h) (3y/2) - (5x/3) = -2           (y/3) + (x/3) = 13/6

  (i) x - y = 3                           (x/3) + (y/2) = 6

  (j) (2x/3) + (y/2) = -1             (-x/3) + y = 3

  (k) 5x + 8y = 9                       2x + 3y = 4

  (l) 3 - 2(3a - 4b) = -59            (a - 3)/4 - (b - 4)/5 = 2¹/₁₀

Answers for the worksheet on simultaneous linear equations are given below to check the exact answers of the above questions on system of linear equations.

1. (a) x = 9, y = 6

(b) x = 1, y = -1

(c) x = 1, y = -1

(d) x = 261/49, y = 9/49

(e) x = -8/23, y = -21/23

(f) x = 4, y = 2

(g) x = 6, y = 1

(h) x = -3, y = 2

(i) x = 4, y = 3

2. (a) x = -2, y = -1

(b) x = -1, y = 1

(c) x = 2, y = 1

(d) x = -1/2, y = 3

(e) x = 3, y = 1

(f) x = 1/2, y = 2

(g) x = -1, y = 1/3

(h) x = 1, y = 2

(i) x = -21/23, y = 26/23

3. (a) a = 5, b = 7

(b) x = 5, y = -1

(c) x = -1, y = 1/3

(d) p = 5, q = 6

(e) l = -2, m = -65

(f) x = 2, y = 1

(g) x = 2, y = 3

(h) x = 141/38, y = 53/19

(i) x = 9, y = 6

(j) x = -3, y = 2

(k) x = 5, y = -2

(l) a = 5, b = -4

●   Simultaneous Linear Equations

Simultaneous Linear Equations

Comparison Method

Elimination Method

Substitution Method

Cross-Multiplication Method

Solvability of Linear Simultaneous Equations

Pairs of Equations

Word Problems on Simultaneous Linear Equations

Practice Test on Word Problems Involving Simultaneous Linear Equations

●   Simultaneous Linear Equations - Worksheets

Worksheet on Problems on Simultaneous Linear Equations

8th Grade Math Practice From Worksheet on Simultaneous Linear Equations to HOME PAGE

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Worded simultaneous equations problems

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

18 May 2015

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Really nice differentiated questions I can use as consolidation and extension work. I noticed a mistake in the answers for question 3B: the area is 91 not 30.

Empty reply does not make any sense for the end user

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Great stuff

muneebabrar84

How do you do question 6 in column B???

Fully worked answers rather than just the numerical answers would help the student in understanding HOW to approach these

I like it, but question 7 has an error. The sum of the square of the two shortest sides equals 400. Q6 was beyond me without trial and error.

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