More simultaneous equations - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

More simultaneous equations

Equations and inequalities

Lesson 6 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Equations and inequalities

Lesson 6 of 7

Answer each one, then check.

1

Solve x + y = 5 and x − y = 1.

2

Multiply 2x + 3y = 5 by 3.

3

What is the lowest common multiple of 2 and 3?

4

Solve 4y = 12.

5

What is −6 + 6?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Equations and inequalities

Lesson 6 of 7

1

Solve x + y = 5 and x − y = 1.

x = 3, y = 2

2

Multiply 2x + 3y = 5 by 3.

6x + 9y = 15

3

What is the lowest common multiple of 2 and 3?

6

4

Solve 4y = 12.

y = 3

5

What is −6 + 6?

0

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Equations and inequalities

Lesson 6 of 7

1

Multiply one equation to match coefficients.

2

Multiply both equations to match coefficients.

3

Solve worded problems using two equations.

4

Check that answers satisfy both equations.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Matching the Coefficients

Equations and inequalities

Lesson 6 of 7

Multiply so that one pair of coefficients matches, then add or subtract.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Multiplying Before Eliminating

Equations and inequalities

Lesson 6 of 7

When no pair of coefficients matches.

1

Choose the unknown to eliminate

Look for the easiest pair of coefficients

2

Find the lowest common multiple

For 3 and 2 it is 6

3

Multiply each equation

So the coefficients of that unknown match

4

Add or subtract

Same signs subtract, opposite signs add

5

Solve, substitute, check

Two values, checked in both equations

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Multiplying Both Equations

Equations and inequalities

Lesson 6 of 7

Solve 2x + 3y = 13 and 5x − 2y = 4.

1

Multiply (1) by 2 and (2) by 3 to match the y terms

4x + 6y = 26 and 15x − 6y = 12

2

The y terms have opposite signs, so add

19x = 38

3

Solve

x = 2

4

Substitute into (1)

4 + 3y = 13, so y = 3

5

Check in (2)

5 × 2 − 2 × 3 = 4

ANSWER

x = 2 and y = 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Another Pair

Equations and inequalities

Lesson 6 of 7

Solve 3x + 2y = 12 and 5x − 3y = 1.

1

Multiply (1) by 3 and (2) by 2

9x + 6y = 36 and 10x − 6y = 2

2

Add

19x = 38, so x = 2

3

Substitute into (1)

6 + 2y = 12, so y = 3

4

Check in (2)

10 − 9 = 1

ANSWER

x = 2 and y = 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Number Problem

Equations and inequalities

Lesson 6 of 7

The sum of two numbers is 25. Twice the first number plus three times the second number is 62. Find the numbers.

1

Let the numbers be x and y

x + y = 25 and 2x + 3y = 62

2

Multiply the first by 2

2x + 2y = 50

3

Subtract from the second

y = 12

4

Find x

x = 25 − 12 = 13

ANSWER

The numbers are 13 and 12.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Cost Problem

Equations and inequalities

Lesson 6 of 7

3 apples and 2 bananas cost £1.80. 2 apples and 5 bananas cost £2.30. Find the cost of one apple and one banana.

1

Write the equations in pence

3a + 2b = 180 and 2a + 5b = 230

2

Multiply (1) by 2 and (2) by 3

6a + 4b = 360 and 6a + 15b = 690

3

Subtract

11b = 330, so b = 30

4

Find a

3a + 60 = 180, so a = 40

ANSWER

An apple costs 40p and a banana costs 30p.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Common Slips

Equations and inequalities

Lesson 6 of 7

Check every step.

Multiply every term

Both sides and every term of the equation.

Signs when subtracting

Subtracting −6y adds 6y.

Wrong equation for substitution

Substitute into either original equation, then check in the other.

Units

In a money problem, keep everything in pence or everything in pounds.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Equations and inequalities

Lesson 6 of 7

Simultaneous equations

Equations that must both be true for the same unknown values.

Elimination

Removing an unknown by adding or subtracting.

Lowest common multiple

The smallest number that is a multiple of two numbers.

Coefficient

The number multiplying an unknown.

Substitute

Replace an unknown by its value.

Check

Put the answers back into both equations.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and inequalities

Lesson 6 of 7

YOUR TASK

Spot the Method

12 minutes

For each pair, say how you would eliminate an unknown, then solve. (a) x + 2y = 8 and 3x − y = 3 (b) 3x + 4y = 25 and 5x − 2y = 7 (c) 2x + 5y = 16 and 3x − 2y = 5.

1

Choose the unknown to eliminate.

2

Multiply to match.

3

Solve and check.

WHAT A GOOD ANSWER SHOWS

(a) Multiply the second by 2: 6x − 2y = 6, add: 7x = 14, so x = 2, y = 3. (b) Multiply the second by 2: 10x − 4y = 14, add: 13x = 39, so x = 3, y = 4. (c) Multiply the first by 2 and the second by 5: 4x + 10y = 32 and 15x − 10y = 25, add: 19x = 57, so x = 3, y = 2.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Equations and inequalities

Lesson 6 of 7

Multiply one equation to match coefficients.

Multiply both equations.

Choose add or subtract correctly.

Find both unknowns.

Check in both equations.

Form equations from a worded problem.

Work in consistent units.

Explain each step.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Multiply so that one pair of coefficients matches.

Same signs: subtract. Opposite signs: add.

Substitute back and check in both equations.

In problems, define the letters and write two equations.

EXAM FOCUS

Solve the simultaneous equations 2x + 3y = 13 and 5x − 2y = 4. (4 marks)

Multiply both equations so that the y-coefficients match (6 and 6), then add because the signs are opposite.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: More Simultaneous Equations

Equations and inequalities

Lesson 6 of 7

Answer all questions. Show your working. · 35 minutes

Question 1 · 4 marks · Non-calculator

Solve the simultaneous equations 2x + 3y = 13 and 5x − 2y = 4.

Question 2 · 4 marks · Non-calculator

Solve the simultaneous equations 3x + 2y = 12 and 5x − 3y = 1.

Question 3 · 4 marks · Non-calculator

The sum of two numbers is 25. Twice the first number plus three times the second number is 62. Work out the two numbers.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: More Simultaneous Equations (cont.)

Equations and inequalities

Lesson 6 of 7

Question 4 · 4 marks · Calculator

3 apples and 2 bananas cost £1.80. 2 apples and 5 bananas cost £2.30. Work out the cost of one apple and the cost of one banana.

Question 5 · 3 marks · Non-calculator

Find the coordinates of the point where the lines y = 3x − 2 and y = x + 4 cross.

Question 6 · 4 marks · Non-calculator

Solve the simultaneous equations 4x + y = 14 and 2x − 3y = 0.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 4 marks · Non-calculator

Equations and inequalities

Lesson 6 of 7

“

Solve the simultaneous equations 2x + 3y = 13 and 5x − 2y = 4.

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 1 · mark scheme

Equations and inequalities

Lesson 6 of 7

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

Multiplying to match coefficients

M1

19x = 38

M1

x = 2

A1

y = 3

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 4 marks · Non-calculator

Equations and inequalities

Lesson 6 of 7

“

Solve the simultaneous equations 3x + 2y = 12 and 5x − 3y = 1.

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 2 · mark scheme

Equations and inequalities

Lesson 6 of 7

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

Multiplying to match coefficients

M1

19x = 38

M1

x = 2

A1

y = 3

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 4 marks · Non-calculator

Equations and inequalities

Lesson 6 of 7

“

The sum of two numbers is 25. Twice the first number plus three times the second number is 62. Work out the two numbers.

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 3 · mark scheme

Equations and inequalities

Lesson 6 of 7

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

Forming both equations

M1

A correct elimination step

M1

y = 12

A1

x = 13

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 4 marks · Calculator

Equations and inequalities

Lesson 6 of 7

“

3 apples and 2 bananas cost £1.80. 2 apples and 5 bananas cost £2.30. Work out the cost of one apple and the cost of one banana.

HOW TO ANSWER IT

Command word: 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 6 of 7

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

Forming both equations

M1

Multiplying to match

M1

b = 30 or a = 40

A1

Both correct with units

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 3 marks · Non-calculator

Equations and inequalities

Lesson 6 of 7

“

Find the coordinates of the point where the lines y = 3x − 2 and y = x + 4 cross.

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 5 · mark scheme

Equations and inequalities

Lesson 6 of 7

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

Equating the two expressions

M1

x = 3

A1

(3, 7)

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 4 marks · Non-calculator

Equations and inequalities

Lesson 6 of 7

“

Solve the simultaneous equations 4x + y = 14 and 2x − 3y = 0.

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 6 · mark scheme

Equations and inequalities

Lesson 6 of 7

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

A method to eliminate or substitute

M1

7y = 14 or 14x = 42

M1

y = 2

A1

x = 3

A1

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