Solving simple simultaneous equations.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving simple simultaneous equations

Equations and inequalities

Lesson 5 of 7

openrevise.com

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.

openrevise.com

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

openrevise.com

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.

openrevise.com

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.

openrevise.com

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.

openrevise.com

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

openrevise.com

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

openrevise.com

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

openrevise.com

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

openrevise.com

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.

openrevise.com

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.

openrevise.com

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.

openrevise.com

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.

openrevise.com

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.

openrevise.com

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.

openrevise.com