Solving simple simultaneous equations - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving simple simultaneous equations

Equations and inequalities

Lesson 5 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Equations and inequalities

Lesson 5 of 7

Answer each one, then check.

1

Solve 2x + 3 = 11.

2

What is 7 − (−3)?

3

Substitute x = 2 into y = 3x − 1.

4

What is the y-intercept of y = 2x + 5?

5

Solve x − 4 = −1.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Equations and inequalities

Lesson 5 of 7

1

Solve 2x + 3 = 11.

x = 4

2

What is 7 − (−3)?

10

3

Substitute x = 2 into y = 3x − 1.

y = 5

4

What is the y-intercept of y = 2x + 5?

5

5

Solve x − 4 = −1.

x = 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Equations and inequalities

Lesson 5 of 7

1

Explain what simultaneous equations are.

2

Solve by elimination when a pair of coefficients match.

3

Solve by substitution.

4

Set up simultaneous equations from a problem and read solutions from graphs.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and inequalities

Lesson 5 of 7

SIMULTANEOUS EQUATIONS

Simultaneous equations have two unknowns, and the solution is the pair of values that makes both equations true at the same time.

On a graph, the solution is where the two lines cross.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Where Two Lines Meet

Equations and inequalities

Lesson 5 of 7

The crossing point is the only pair of values on both lines.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving by Elimination

Equations and inequalities

Lesson 5 of 7

Make one unknown disappear.

1

Label the equations

Call them (1) and (2)

2

Match the coefficients

If they are already equal or opposite, go on; if not, multiply

3

Add or subtract

Add if the signs are opposite; subtract if they are the same

4

Solve for one unknown

You now have a one-variable equation

5

Substitute back

Find the other unknown, then check in the other equation

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Elimination by Adding

Equations and inequalities

Lesson 5 of 7

Solve x + y = 7 and x − y = 1.

1

The y terms have opposite signs, so add

(x + y) + (x − y) = 7 + 1

2

Simplify

2x = 8, so x = 4

3

Substitute into x + y = 7

4 + y = 7, so y = 3

4

Check in x − y = 1

4 − 3 = 1

ANSWER

x = 4 and y = 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Elimination by Subtracting

Equations and inequalities

Lesson 5 of 7

Solve 2x + y = 11 and x + y = 7.

1

The y terms have the same sign, so subtract

(2x + y) − (x + y) = 11 − 7

2

Simplify

x = 4

3

Substitute into x + y = 7

y = 3

4

Check

2 × 4 + 3 = 11

ANSWER

x = 4 and y = 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving by Substitution

Equations and inequalities

Lesson 5 of 7

Solve y = 2x − 1 and x + y = 8.

1

Substitute y = 2x − 1 into the second equation

x + (2x − 1) = 8

2

Simplify

3x − 1 = 8, so 3x = 9 and x = 3

3

Find y

y = 2 × 3 − 1 = 5

ANSWER

x = 3 and y = 5

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Word Problem

Equations and inequalities

Lesson 5 of 7

2 adult tickets and 3 child tickets cost £27. 1 adult ticket and 2 child tickets cost £15. Find the cost of each ticket.

1

Let a be an adult ticket and c a child ticket

2a + 3c = 27 and a + 2c = 15

2

Rearrange the second

a = 15 − 2c

3

Substitute

2(15 − 2c) + 3c = 27, so 30 − 4c + 3c = 27

4

Solve

c = 3, so a = 15 − 6 = 9

ANSWER

An adult ticket costs £9 and a child ticket costs £3.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Elimination or Substitution?

Equations and inequalities

Lesson 5 of 7

ELIMINATION

• Best when the equations are both in the form ax + by = c.

• Add or subtract to remove one unknown.

• Needs matching coefficients.

SUBSTITUTION

• Best when one equation already gives y = … or x = ….

• Replace the unknown in the other equation.

• Watch brackets and signs.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Equations and inequalities

Lesson 5 of 7

Simultaneous equations

Two or more equations that are true at the same time.

Unknown

A letter standing for a number to be found.

Elimination

Removing one unknown by adding or subtracting equations.

Substitution

Replacing an unknown with an expression.

Solution

The pair of values that satisfies both equations.

Coefficient

The number multiplying an unknown.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and inequalities

Lesson 5 of 7

YOUR TASK

Solve and Check

12 minutes

Solve each pair of simultaneous equations, then check your answer in both equations. (a) x + y = 10, x − y = 4 (b) 3x + y = 14, x + y = 8 (c) y = x + 2, 2x + y = 11.

1

Choose elimination or substitution.

2

Solve.

3

Check in both equations.

WHAT A GOOD ANSWER SHOWS

(a) x = 7, y = 3. (b) 2x = 6, so x = 3, y = 5. (c) 2x + x + 2 = 11, so x = 3, y = 5.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Equations and inequalities

Lesson 5 of 7

Explain what simultaneous equations are.

Eliminate by adding.

Eliminate by subtracting.

Substitute one equation into the other.

Find both unknowns.

Check my solution.

Set up equations from a problem.

Read the solution from a graph.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Same signs: subtract. Opposite signs: add.

Substitute back to find the second unknown.

Always check in both equations.

On a graph, the solution is the crossing point.

EXAM FOCUS

2 adult tickets and 3 child tickets cost £27. 1 adult ticket and 2 child tickets cost £15. Work out the cost of an adult ticket and the cost of a child ticket. (4 marks)

Define letters, write two equations, then solve. Check that both original equations work with your answer.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Solving Simple Simultaneous Equations

Equations and inequalities

Lesson 5 of 7

Answer all questions. Show your working. · 30 minutes

Question 1 · 3 marks · Non-calculator

Solve the simultaneous equations x + y = 7 and x − y = 1.

Question 2 · 3 marks · Non-calculator

Solve the simultaneous equations 2x + y = 11 and x + y = 7.

Question 3 · 3 marks · Non-calculator

Solve the simultaneous equations y = 2x − 1 and x + y = 8.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Solving Simple Simultaneous Equations (cont.)

Equations and inequalities

Lesson 5 of 7

Question 4 · 4 marks · Non-calculator

2 adult tickets and 3 child tickets cost £27 in total. 1 adult ticket and 2 child tickets cost £15 in total. Work out the cost of an adult…

Question 5 · 2 marks · Non-calculator

The graph shows the lines y = 2x − 3 and x + y = 6. Use the graph to solve the simultaneous equations y = 2x − 3 and x + y = 6.

Question 6 · 3 marks · Non-calculator

Solve the simultaneous equations 3x + 2y = 16 and x − y = 2.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 3 marks · Non-calculator

Equations and inequalities

Lesson 5 of 7

“

Solve the simultaneous equations x + y = 7 and x − y = 1.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Equations and inequalities

Lesson 5 of 7

3 marks available. Award a mark for each point made.

Adding or a correct elimination

M1

x = 4

A1

y = 3

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 3 marks · Non-calculator

Equations and inequalities

Lesson 5 of 7

“

Solve the simultaneous equations 2x + y = 11 and x + y = 7.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Equations and inequalities

Lesson 5 of 7

3 marks available. Award a mark for each point made.

Subtracting the equations

M1

x = 4

A1

y = 3

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 3 marks · Non-calculator

Equations and inequalities

Lesson 5 of 7

“

Solve the simultaneous equations y = 2x − 1 and x + y = 8.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Equations and inequalities

Lesson 5 of 7

3 marks available. Award a mark for each point made.

Substituting

M1

x = 3

A1

y = 5

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 4 marks · Non-calculator

Equations and inequalities

Lesson 5 of 7

“

2 adult tickets and 3 child tickets cost £27 in total. 1 adult ticket and 2 child tickets cost £15 in total. Work out the cost of an adult ticket and the cost of a child ticket.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 4 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Equations and inequalities

Lesson 5 of 7

4 marks available. Award a mark for each point made.

Forming both equations

M1

A correct elimination or substitution

M1

One value correct

A1

Both values with units

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 2 marks · Non-calculator

Equations and inequalities

Lesson 5 of 7

The graph shows the lines y = 2x − 3 and x + y = 6. Use the graph to solve the simultaneous equations y = 2x − 3 and x + y = 6. (2 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Equations and inequalities

Lesson 5 of 7

2 marks available. Award a mark for each point made.

Reading the crossing point

M1

x = 3 and y = 3

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 3 marks · Non-calculator

Equations and inequalities

Lesson 5 of 7

“

Solve the simultaneous equations 3x + 2y = 16 and x − y = 2.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Equations and inequalities

Lesson 5 of 7

3 marks available. Award a mark for each point made.

Substituting or multiplying to match

M1

y = 2

A1

x = 4

A1

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