Substitute means to put in place of another. In algebra, substitution can be used to replace variables with values, terms, or expressions.

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

Problem solving with simultaneous equations

Maths

In the future unit, pupils will learn how to extend their understanding of solving equations and extend this to consider inequalities. Pupils will consider a variety of inequalities and ways that their solutions can be represented.

In the prior unit, pupils learnt to solve quadratic equations by factorising and this skill will be required in this unit through manipulating two simultaneous equations to find the solutions. For pupils studying the higher content, solving quadratic equations is required if one of the simultaneous equations is a quadratic.

Simultaneous equations: 2 variables

Which two equations represent this scenario: Nine small bags (s) and five bigger bags (b) contains 182 counters in total.
Two small bags and four bigger bags contain 104 counters in total.

Which two equations represent this scenario: Five small bags (s) and three bigger bags (b) contains 94 counters in total. Seven small bags and four bigger bags contain 128 counters in total.

A bakery sells cookies in boxes of small and large sizes. Four small and three large boxes contain 235 cookies. Two small boxes and five large boxes contain 275 cookies. How many are in each box?

A farmer sells bags of apples and oranges. Three bags of apples and four bags of oranges weigh 73 kg. Five bags of apples and three bags of oranges weigh 85 kg. What is the weight of each bag?

A company produces two types of gadget: standard (s) and premium (p). Selling 5 s and 3 p gadgets makes £260. Selling 3 s and 5 p gadgets makes £220. How much does each type of gadget cost?

A company produces two types of gadget: standard (s) and premium (p). Selling 6 s and 4 p gadget makes £560. Selling 4 s and 6 p gadgets makes £640. How much does each type of gadget cost?

Solve using any method: $$5x + 4y = 11$$ and $$2x + 5y = 35$$. Write your answer in the form $$(x,y)$$.

Solve using any method: $$x + 5y =24$$ and $$10x + 6y = 20$$. Write your answer in the form $$(x,y)$$.

Solve using any method: $$8x + 2y =10$$ and $$6x - 8y = 36$$. Write your answer in the form $$(x,y)$$.

Solve using any method: $$12x + 4y = 60$$ and $$12x - 10y =102$$. Write your answer in the form $$(x,y)$$.

Solve using any method: $$3x + y = 15$$ and $$10x - 10y = 90$$. Write your answer in the form $$(x,y)$$.

Solve using any method: $$3x + 5y = -6$$ and $$10x - 10y = 60$$. Write your answer in the form $$(x,y)$$.

This lesson will be removed by end of Summer Term 2025.

Problem solving with simultaneous equations

Lesson outcome

This lesson will be removed by end of Summer Term 2025.