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Combining equations

I can additively combine two equations to create a third valid equation.

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Lesson 3 of 13

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Year 10

Combining equations

I can additively combine two equations to create a third valid equation.

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Lesson details

Key learning points

Two equations can be combined into a third equation.
You can combine equations using the standard arithmetic operations.
The new equation may not contain both variables any more.

Keywords

Elimination - Elimination is a technique to help solve equations simultaneously and is where one of the variables in a problem is removed.

Common misconception

Mistakes can be made when subtracting negative values. E.g. thinking that 5 - - 2 = 3.

Consider additive inverses, subtracting a negative is the same as adding the additive inverse. Pupils should be able to spot mistakes in learning cycle 1 by checking their new equation is still true.

Pupils should get used to labelling their equations so they can make it clear which equations they are manipulating and what operations they are performing when they are writing their solutions.

Teacher tip

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
Which pair of coordinates does not satisfy the equation $$y = 2x + 4$$?

(1, 6)

(2, 8)

(3, 10)

Correct answer: (4, 11)

Q2.
The area of a rectangle is $$90 cm^2$$. The perimeter is $$94 cm$$. Which two equations are true for side lengths $$a$$ and $$b$$?

Correct answer: $$ab = 90$$

$$a + b = 94$$

Correct answer: $$2(a + b) = 94$$

$$ab = 94$$

Q3.
Which two of the following are correct for a right triangle with sides $$a, b$$ and hypotenuse $$c$$?

$$a + b + c = abc$$

Correct answer: $$Area = \frac{ab}{2}$$

Correct answer: $$Perimeter = a + b + c$$

$$Area = ab$$

Q4.
If $$x = -9$$, what is the value of $$y$$ for $$x + 2y = 13$$?

Correct Answer: 11, y=11

Q5.
Which pair of coordinates does not satisfy the equation $$y = -3x + 2$$?

(-1, 5)

(0, 2)

Correct answer: (1, 1)

(2, -4)

Q6.
There are 12 goals scored in a football match. Team A scored $$a$$ goals, and winning Team B scored $$b$$ goals. The winning team scored 5 more than the losing team.

Correct answer: $$a + b = 12$$

$$a + b = 5$$

$$a = b - 5$$

Correct answer: $$a = b - 5$$

Assessment exit quiz

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6 Questions

Q1.
Which equation is true for $$x = 2$$ and $$y = 5$$?

$$2x + y = -8$$

Correct answer: $$2x + 3y = 19$$

$$4x + 2y = 20$$

Q2.
Which equation is true for $$x = 3$$ and $$y = 9$$?

$$2x + 3y = 12$$

$$3x + y = 16$$

Correct answer: $$2(x + y) = 24$$

Q3.
Which equation is true for $$x = -2$$ and $$y = 3$$?

Correct answer: $$3y - x = 11$$

$$3y - x = 7$$

$$3x - y = 9$$

Q4.
Which equation is true for $$x = 8$$ and $$y = 1$$?

$$x^2 - y^2 = 64$$

$$x^2 = 34y$$

Correct answer: $$x^2 + y = 65$$

Q5.
If I add together the pair of simultaneous equations $$5x + 7y = 87$$ and $$7x + 7y = 91$$, what is the resulting equation?

Correct Answer: 12x + 14y = 178

Q6.
If I add together the pair of simultaneous equations $$5x + 2y = 10$$ and $$7x - 2y = -50$$, what is the resulting equation?

Correct Answer: 12x = -40

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UnitMaths / Simultaneous equations: 2 variables

Maths

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Combining equations

Combining equations

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.Which pair of coordinates does not satisfy the equation $$y = 2x + 4$$?

Q2.The area of a rectangle is $$90 cm^2$$. The perimeter is $$94 cm$$. Which two equations are true for side lengths $$a$$ and $$b$$?

Q3.Which two of the following are correct for a right triangle with sides $$a, b$$ and hypotenuse $$c$$?

Q4.If $$x = -9$$, what is the value of $$y$$ for $$x + 2y = 13$$?

Q5.Which pair of coordinates does not satisfy the equation $$y = -3x + 2$$?

Q6.There are 12 goals scored in a football match. Team A scored $$a$$ goals, and winning Team B scored $$b$$ goals. The winning team scored 5 more than the losing team.

Assessment exit quiz

6 Questions

Q1.Which equation is true for $$x = 2$$ and $$y = 5$$?

Q2.Which equation is true for $$x = 3$$ and $$y = 9$$?

Q3.Which equation is true for $$x = -2$$ and $$y = 3$$?

Q4.Which equation is true for $$x = 8$$ and $$y = 1$$?

Q5.If I add together the pair of simultaneous equations $$5x + 7y = 87$$ and $$7x + 7y = 91$$, what is the resulting equation?

Q6.If I add together the pair of simultaneous equations $$5x + 2y = 10$$ and $$7x - 2y = -50$$, what is the resulting equation?

Lesson appears in

UnitMaths / Simultaneous equations: 2 variables

Maths

Q1.
Which pair of coordinates does not satisfy the equation $$y = 2x + 4$$?

Q2.
The area of a rectangle is $$90 cm^2$$. The perimeter is $$94 cm$$. Which two equations are true for side lengths $$a$$ and $$b$$?

Q3.
Which two of the following are correct for a right triangle with sides $$a, b$$ and hypotenuse $$c$$?

Q4.
If $$x = -9$$, what is the value of $$y$$ for $$x + 2y = 13$$?

Q5.
Which pair of coordinates does not satisfy the equation $$y = -3x + 2$$?

Q6.
There are 12 goals scored in a football match. Team A scored $$a$$ goals, and winning Team B scored $$b$$ goals. The winning team scored 5 more than the losing team.

Q1.
Which equation is true for $$x = 2$$ and $$y = 5$$?

Q2.
Which equation is true for $$x = 3$$ and $$y = 9$$?

Q3.
Which equation is true for $$x = -2$$ and $$y = 3$$?

Q4.
Which equation is true for $$x = 8$$ and $$y = 1$$?

Q5.
If I add together the pair of simultaneous equations $$5x + 7y = 87$$ and $$7x + 7y = 91$$, what is the resulting equation?

Q6.
If I add together the pair of simultaneous equations $$5x + 2y = 10$$ and $$7x - 2y = -50$$, what is the resulting equation?