Simultaneous equations (2022)

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 e.g. x and y.

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

For example, below are some simultaneous equations:

2x + 4y = 14 
4x − 4y = 4 
6a + b = 18 
4a + b = 14 
3h + 2i = 8 
2h + 5i = −2 

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.

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.

E.g.

\[\begin{aligned}x+y=6\\-3x+y&=2\end{aligned}\]

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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.

E.g.

\[2x + 5=0\]

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.

E.g.

\[x^{2}-2x+1=0\]

Quadratic simultaneous equations are solved by the substitution method.

See also: 15 Simultaneous equations questions

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Simultaneous equations worksheets

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

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Simultaneous equations worksheets

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

DOWNLOAD FREE

How to solve simultaneous equations

To solve pairs of simultaneous equations you need to:

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

How do you solve pairs of simultaneous equations?

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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 to2e.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

(Video) Simultaneous Equations Math Lesson

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)

Solve:

\[\begin{aligned}2x+4y&=14\\4x-4y&=4\end{aligned}\]

  1. Eliminate one of the variables.

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

\[\begin{aligned}2x+4y&=14\\4x-4y&=4\\\hline6x&=18\end{aligned}\]

2Find the value of one variable.

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3Find the value of the remaining variable via substitution.

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

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4Clearly state the final answer.

\[x=3 \qquad\qquad y=2\]

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

\[\begin{aligned}4(3)+4(2)&=4\\12-8&=4\\\end{aligned}\]

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.

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Example 2: Solving simultaneous equations by elimination (subtraction)

Solve:

\[\begin{array}{l}6 a+b=18 \\4 a+b=14\end{array}\]

  1. Eliminate one of the variables.

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

\[\begin{aligned}6a+b&=18 \\4a+b&=14 \\\hline2a&=4\end{aligned}\]

NOTE: b − b = 0 so b is eliminated

2Find the value of one variable.

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3Find 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.

\[\begin{aligned}6 a+b &=18 \\6(2)+b &=18 \\12+b &=18 \\b &=6\end{aligned}\]

4Clearly state the final answer.

\[a=2 \qquad\qquad b=6\]

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

\[\begin{aligned}4(2)+(6) &=14 \\8+6 &=14\end{aligned}\]

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

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.

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

Solve:

\[\begin{array}{l}3 h+2 i=8 \\2 h+5 i=-2\end{array}\]

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

\[\begin{array}{l}3 h+2 i=8 \\2 h+5 i=-2 \\\hline 5 h+7 i=6\end{array}\begin{aligned}3 h+2 i&=8 \\2 h+5 i&=-2 \\\hline h-3 i&=10\end{aligned}\]

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.

  1. Eliminate one of the variables.

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.

\[\begin{aligned}3h+2 i&=8 \\2h+5 i&=-2 \\\\6h+4 i&=16 \\6h+15 i&=-6\end{aligned}\]

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.

\[\begin{aligned}6 h+4 i&=16 \\6 h+15 i&=-6 \\\hline-11 i&=22\end{aligned}\]

Note: 6h − 6h = 0 so h is eliminated

Careful: 16 − − 6 = 22

2Find the value of one variable.

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

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

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4Clearly state the final answer.

\[h=4 \qquad\qquad i=-2\]

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

\[\begin{aligned}2(4)+5(-2)&=-2 \\8-10&=-2\end{aligned}\]

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

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.

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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.

\[\begin{aligned}10a+6b&=5 \\3a+1b&=1.30\end{aligned}\]

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

(Video) Method of Elimination Steps to Solve Simultaneous Equations

  1. Eliminate one of the variables.

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.

\[\begin{aligned}10 a+6 b&=5 \\3 a+1 b&=1.30 \\\\10 a+6 b&=5 \\18 a+6 b&=7.80\end{aligned}\]

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.

\[\begin{aligned}10a+6b &=5 \\18a+6b &=7.80 \\\hline-8a &=-2.80\end{aligned}\]

NOTE: 6b − 6b = 0 so b is eliminated

16 − − 6 = 22

2Find the value of one variable.

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Note: we ÷ (− 8) not 8

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

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

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4Clearly state the final answer.

\[a=0.35 \qquad\qquad b=0.25\]

So

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

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

\[\begin{aligned}3(0.35)+1(0.25) & =1.30 \\1.05+0.25 & =1.30\end{aligned}\]

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

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.

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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

x=\frac{5}{2}=2.5,\quad y=11

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x=11,\quad y=\frac{5}{2}=2.5

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x=6,\quad y=1

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x=3,\quad y=6

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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 .

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

x=1,\quad y=2

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x=1,\quad y=3

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x=18,\quad y=5

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x=8,\quad y=3

(Video) Simultaneous Equations - Tons of Examples!

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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 .

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

x=4,\quad y=2

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x=4,\quad y=9

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x=3,\quad y=1

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x=3,\quad y=2

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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.

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4. Solve the Simultaneous Equation:

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

x=6,\quad y=2

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x=15,\quad y=4

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x=5,\quad y=9

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x=-6,\quad y=-2

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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.

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Simultaneous equations GCSE questions

1. Solve the simultaneous equations

\begin{array}{l}3 y+x=-4 \\3 y-4 x=6\end{array}

(4 marks)

Show answer

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

(1)

One unknown substituted back into either equation

(Video) Simultaneous Equations (Higher & Foundation) | GCSE Maths Tutor

(1)

y=-\frac{2}{3} \text { oe }

(1)

x=-2

(1)

2. Solve the simultaneous equations

\begin{array}{l}x+3y=12 \\5x-y=4\end{array}

(4 marks)

Show answer

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

\begin{array}{l}5x+15y=60 \\5x-y=4\end{array}

(1)

Or

\begin{array}{l}x+3y=12 \\15x-3y=12\end{array}

(1)

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

(1)

One unknown substituted back into either equation

ft (1)

y=\frac{7}{2} \text { oe }

x=\frac{3}{2} \text { oe }

(1)

3. Solve the simultaneous equations

\begin{array}{l}4x+y=25 \\x-3y=16\end{array}

(4 marks)

Show answer

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

\begin{array}{l}12x+3y=75 \\x-3y=16\end{array}

(1)

Or

\begin{array}{l}4x+y=25 \\4x-12y=64\end{array}

(1)

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

(1)

One unknown substituted back into either equation

ft (1)

x=7 \text { oe }

y=-3 \text { oe }

(1)

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

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Simultaneous equations (38)

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FAQs

What is simultaneous equation with example? ›

Simultaneous equations are two or more algebraic equations with the same unknown variables and the same value of the variables satisfies all such equations. This implies that the simultaneous equations have a common solution. Some of the examples of simultaneous equations are: 2x - 4y = 4, 5x + 8y = 3.

What is the rule for simultaneous equations? ›

If the signs are different, add the equations together. If the signs are the same, subtract them.

What do you mean by simultaneous equation? ›

a set of two or more equations, each containing two or more variables whose values can simultaneously satisfy both or all the equations in the set, the number of variables being equal to or less than the number of equations in the set.

How do you solve two equations with simultaneous equations? ›

What is the fastest way to solve simultaneous equations? ›

Trick To Solving Simultaneous Equations Easily - YouTube

How many types of simultaneous equations are there? ›

There are three different approaches to solve the simultaneous equations such as substitution, elimination, and augmented matrix method.

Are simultaneous equations hard? ›

Simultaneous equations can appear quite difficult at first, but once you've practised the methods it should get much easier.

How do you rearrange and solve simultaneous equations? ›

Solving simultaneous equations by rearranging before elimination

How do you solve 4 simultaneous equations? ›

solving system of 4 variables (elimination) - YouTube

Why are simultaneous equations important? ›

Simultaneous equations can be used when considering the relationship between the price of a commodity and the quantities of the commodity people want to buy at a certain price. An equation can be written that describes the relationship between quantity, price and other variables, such as income.

What are the characteristics of simultaneous equation? ›

The following characteristics are attributed to ratio relationship: i) Ratio is a cross relation found between two or more quantities of same type. ii) It must be expressed in the same units. iv) A ratio expresses the number of times that one quantity contains another.

What is the difference between linear equation and simultaneous equations? ›

A linear equation is an equation in which each term is either constant or the product of a constant. It is a one-degree equation. Simultaneous equations are a set of equations containing more than one variables.

How do you solve hard simultaneous equations? ›

Simultaneous equations GCSE maths - grade 7+ quite difficult 2 - YouTube

How do you do simultaneous? ›

Simultaneous Equations - Example to solve 3 - YouTube

How do you solve simultaneous equations without a calculator? ›

Simultaneous Equations - Algebra Math Trick - YouTube

What is substitution method in simultaneous equation? ›

Step 1: Solve one of the equations for one of the variables. Step 2: Substitute that expression into the remaining equation. The result will be a linear equation with one variable that can be solved. Step 3: Solve the remaining equation.

How do you solve 3 simultaneous equations? ›

Pick any two pairs of equations from the system. Eliminate the same variable from each pair using the Addition/Subtraction method. Solve the system of the two new equations using the Addition/Subtraction method. Substitute the solution back into one of the original equations and solve for the third variable.

How do you do simultaneous equations in two variables? ›

Simultaneous linear (systems of) equations in two variables - 1

What grade level is simultaneous equations? ›

Grade 8 » Expressions & Equations » Analyze and solve linear equations and pairs of simultaneous linear equations. » 8. Analyze and solve pairs of simultaneous linear equations.

Can you divide 2 simultaneous equations? ›

It works because of two properties of equations: Multiplying (or dividing) the expression on each side by the same number does not alter the equation. Adding two equations produces another valid equation: e.g. 2x = x + 10 (x = 10) and x − 3 = 7 (x also = 10).

How do you use graphical method to solve simultaneous equations? ›

Solving Simultaneous Equations Graphically - Corbettmaths - YouTube

How do you solve a linear pair of equations? ›

The steps to solve a pair of linear equations in two equations are:
  1. First, you need to frame the two linear equation in two different variables.
  2. You can solve them either using the elimination, the substitution or the cross multiplication method.
  3. You can also solve them in a graphical manner.

How do you solve an equation with 5 variables? ›

Can you solve 5 linear equations under 5 seconds? - YouTube

How do you solve 3 equations with 3 variables? ›

Solving 3 Equations with 3 Unknowns - YouTube

How do you solve 2 equations with 3 variables? ›

Solving a System of 2 Equations with 3 Unknowns - YouTube

How equations are used in real life? ›

Applications of Linear Equations in Real life

It can be used to solve age related problems. It is used to calculate speed, distance and time of a moving object. Geometry related problems can be solved. It is used to calculate money and percentage related problems.

What are some real life examples of equations? ›

Real life examples include:
  • Calculating wages based on an hourly pay rate.
  • Calculating medicine doses based on patients' weights.
  • Calculating the perimeters of squares.
  • Hiring a car if a deposit is paid and there is an hourly charge.

Who invented simultaneous equations? ›

The algorithm of elimination for solving a system of simultaneous linear equations is not difficult and has been discovered more than once. The first time was by the ancient Chinese and appears in chapter 8 of the work Jiuzhang suanshu (Nine Chapters of the Mathematical Art, circa 100 B.C.E. –50 C.E.).

What is simultaneous relationship? ›

A simultaneous relation is a temporal relation in which the events or states of proposition(s) are communicated as occurring at the same time.

What are the assumptions of simultaneous equation Modelling? ›

Simultaneous equation models assume that the endogenous and exogenous variables are directly measured and have no measurement error. The disturbances include all vari- ables influencing y that are omitted from the equation and are assumed to have expected values of zero (E() = 0).

What is simultaneous equation system how would you identify it? ›

A simultaneous equations system is defined as a system with two or more equations, where a variable explained in one equation appears as an explanatory variable in another. Thus, the endogenous variables in the system are simultaneously determined.

How do you determine if an equation is linear or nonlinear? ›

Using an Equation

Simplify the equation as closely as possible to the form of y = mx + b. Check to see if your equation has exponents. If it has exponents, it is nonlinear. If your equation has no exponents, it is linear.

How do you add two equations? ›

Solving Systems of Equations: The Addition Method - YouTube

What is linear simultaneous equations? ›

Two or more linear equations that all contain the same unknown variables are called a system of simultaneous linear equations. Solving such a system means finding values for the unknown variables which satisfy all the equations at the same time.

How do you subtract two equations? ›

Solving system of equation using elimination method - YouTube

Videos

1. The Maths Prof: Solve Simultaneous Equations part 1 (by elimination)
(The Maths Prof)
2. Simultaneous Equations elimination - Corbettmaths
(corbettmaths)
3. Solving simultaneous equations by substitution
(Whiteboard Maths)
4. Olympiad Mathematics | Learn How to Solve the System of Equations Fast | Math Olympiad Preparation
(PreMath)
5. SIMULTANEOUS EQUATIONS FROM WORD PROBLEMS
(Shifting Grades)
6. If the system of equations 3x+y=1 and (2k-1)x+(k-1)y=2k+1 is inconsistent, then k=...
(Math By CV Sir)

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