Equations - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations

Algebra

Lesson 3 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Algebra

Lesson 3 of 7

Answer each one, then check.

1

Solve x + 7 = 12.

2

What is the inverse of "multiply by 4"?

3

Work out −6 ÷ 2.

4

Last lesson: expand 3(x − 2).

5

Last lesson: factorise 6x + 9.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Algebra

Lesson 3 of 7

1

Solve x + 7 = 12.

x = 5

2

What is the inverse of "multiply by 4"?

Divide by 4.

3

Work out −6 ÷ 2.

−3

4

Last lesson: expand 3(x − 2).

3x − 6

5

Last lesson: factorise 6x + 9.

3(2x + 3)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Algebra

Lesson 3 of 7

1

Solve linear equations with the unknown on both sides.

2

Solve equations with brackets.

3

Solve equations with fractions.

4

Form equations from words and diagrams.

5

Solve problems by forming and solving an equation.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Keeping the Balance

Algebra

Lesson 3 of 7

An equation says two things are equal. Solving it means finding the value of the letter that makes it true.

Same on both sides

Add, subtract, multiply or divide both sides by the same thing, and the equation stays true.

Inverse operations

Undo + with −, and × with ÷.

One step at a time

Write each step on a new line, with the = signs lined up.

Check

Put your answer back into the original equation. Both sides should give the same number.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

An Equation Is a Balance

Algebra

Lesson 3 of 7

3x + 2 = 11: take 2 from both sides, then divide by 3.

The scales stay level as long as you do the same thing to both pans. Take 2 counters off each side and 3 bags balance 9 counters; share them out and each bag must hold 3. That is exactly how you solve 3x + 2 = 11.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Method for Any Linear Equation

Algebra

Lesson 3 of 7

Not every equation needs every step - skip the ones that do not apply.

1

Brackets

Expand any brackets.

2

Fractions

Multiply both sides to clear any fractions.

3

Letters

Collect the letter terms on the side with more of them.

4

Numbers

Collect the numbers on the other side.

5

Divide

Divide by the number in front of the letter.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Unknown on Both Sides

Algebra

Lesson 3 of 7

Solve 5x − 7 = 2x + 11

1

Subtract 2x from both sides (there are more xs on the left)

3x − 7 = 11

2

Add 7 to both sides

3x = 18

3

Divide both sides by 3

x = 6

4

Check: both sides give 23

5(6) − 7 = 23 and 2(6) + 11 = 23

ANSWER

x = 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations with Brackets

Algebra

Lesson 3 of 7

Solve 3(2x − 1) = 4(x + 5)

1

Expand both brackets

6x − 3 = 4x + 20

2

Subtract 4x from both sides

2x − 3 = 20

3

Add 3 to both sides

2x = 23

4

Divide by 2

x = 11.5

ANSWER

x = 11.5 (or 23/2)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations with Fractions

Algebra

Lesson 3 of 7

Solve (2x + 1)/3 = (x − 2)/2

1

Multiply both sides by 3 and by 2 (by 6)

2(2x + 1) = 3(x − 2)

2

Expand

4x + 2 = 3x − 6

3

Subtract 3x

x + 2 = −6

4

Subtract 2

x = −8

5

Check: both sides give −5

(−16 + 1)/3 = −5 and (−8 − 2)/2 = −5

ANSWER

x = −8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Two Fractions on One Side

Algebra

Lesson 3 of 7

Solve x/4 + x/6 = 5

1

The lowest common multiple of 4 and 6 is 12: multiply every term by 12

3x + 2x = 60

2

Collect like terms

5x = 60

3

Divide by 5

x = 12

ANSWER

x = 12

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

PART TWO

Forming Equations

Turning words and diagrams into algebra.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

From Words to an Equation

Algebra

Lesson 3 of 7

Four steps turn a problem into an equation you can solve.

Choose a letter

Let x be the thing you do not know - say exactly what it stands for.

Write expressions

Write everything else in terms of x.

Make an equation

Use the fact the question gives you: a total, a perimeter, angles adding to 180°.

Answer the question

Solve for x, then use it to find what the question actually asked for.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Perimeter Problem

Algebra

Lesson 3 of 7

A rectangle has length (3x + 2) cm and width (x − 1) cm. Its perimeter is 42 cm. Find the length and the width.

1

Perimeter = 2 lengths + 2 widths

2(3x + 2) + 2(x − 1) = 42

2

Expand

6x + 4 + 2x − 2 = 42

3

Simplify

8x + 2 = 42

4

Solve

8x = 40, so x = 5

5

Answer the question

Length = 3(5) + 2 = 17, width = 5 − 1 = 4

ANSWER

Length 17 cm, width 4 cm (check: 2 × 17 + 2 × 4 = 42)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Word Problem

Algebra

Lesson 3 of 7

An adult cinema ticket costs £4 more than a child ticket. Two adult tickets and three child tickets cost £38. Find the price of each ticket.

1

Let a child ticket cost £x

An adult ticket costs £(x + 4)

2

Write the equation

2(x + 4) + 3x = 38

3

Expand and simplify

2x + 8 + 3x = 38, so 5x + 8 = 38

4

Solve

5x = 30, so x = 6

ANSWER

Child £6, adult £10 (check: 2 × 10 + 3 × 6 = 38)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Algebra

Lesson 3 of 7

Equation

A statement that two expressions are equal, true for particular values of the letter.

Solve

Find the value of the letter that makes the equation true.

Solution

The value that makes the equation true.

Inverse operation

The operation that undoes another: − undoes +, ÷ undoes ×.

Linear equation

An equation where the highest power of the letter is 1.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Algebra

Lesson 3 of 7

YOUR TASK

Equation Relay

15 minutes

Solve each equation; each answer is used in the next. (a) 4x + 3 = 23. (b) Let a be your answer to (a): solve 2y − a = y + 1. (c) Let b be your answer to (b): solve 3(z − 2) = b + z. (d) Let c be your answer to (c): solve w/2 + w/3 = c + 1.

1

Solve each equation in turn.

2

Check each answer before passing it on.

3

The first team with a correct (d) wins.

WHAT A GOOD ANSWER SHOWS

(a) x = 5 (b) y = 6 (c) 3z − 6 = 6 + z, so z = 6 (d) 5w/6 = 7, so w = 8.4

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Algebra

Lesson 3 of 7

Solve two-step equations.

Solve equations with the unknown on both sides.

Solve equations with brackets.

Solve equations with one fraction.

Solve equations with two fractions.

Check a solution by substituting.

Form an equation from words.

Form an equation from a diagram.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Do the same to both sides to keep the equation balanced.

Expand brackets, clear fractions, collect letters, collect numbers, divide.

Clear fractions by multiplying every term by the LCM of the denominators.

For word problems: choose a letter, write expressions, form an equation, then answer the question asked.

EXAM FOCUS

Solve 4(3 − 2x) = 2(x + 11) (3 marks)

Write each step on a new line. If the final answer is wrong, the method marks for expanding and collecting terms are still there to win.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Equations

Algebra

Lesson 3 of 7

Answer all questions. Show your working. Questions 1 to 4 are non-calculator. · 25 minutes

Question 1 · 2 marks · Non-calculator

Solve 7x + 4 = 3x − 12

Question 2 · 3 marks · Non-calculator

Solve 4(3 − 2x) = 2(x + 11)

Question 3 · 3 marks · Non-calculator

Solve (5x − 2)/3 = (x + 4)/2

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Equations (cont.)

Algebra

Lesson 3 of 7

Question 4 · 4 marks · Non-calculator

The diagram shows a triangle. The sizes of its angles, in degrees, are 2x + 10, 3x − 5 and x + 25. Work out the size of the largest angle.

Question 5 · 4 marks · Calculator

Amy is 3 times as old as Ben. In 8 years' time, Amy will be twice as old as Ben. How old is Amy now?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Non-calculator

Algebra

Lesson 3 of 7

“

Solve 7x + 4 = 3x − 12

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Algebra

Lesson 3 of 7

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

4x = −16, or a correct first step collecting x terms or numbers

M1

x = −4

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 3 marks · Non-calculator

Algebra

Lesson 3 of 7

“

Solve 4(3 − 2x) = 2(x + 11)

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

Algebra

Lesson 3 of 7

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

Both brackets expanded correctly

M1

Correctly collects x terms on one side and numbers on the other

M1

x = −1

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 3 marks · Non-calculator

Algebra

Lesson 3 of 7

“

Solve (5x − 2)/3 = (x + 4)/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 3 · mark scheme

Algebra

Lesson 3 of 7

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

Multiplies both sides to clear the fractions, e.g. 2(5x − 2) = 3(x + 4)

M1

7x = 16

M1

x = 16/7 (or 22/7)

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 4 marks · Non-calculator

Algebra

Lesson 3 of 7

The diagram shows a triangle. The sizes of its angles, in degrees, are 2x + 10, 3x − 5 and x + 25. Work out the size of the largest angle. (4 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Algebra

Lesson 3 of 7

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

Sets the sum of the angles equal to 180

P1

6x + 30 = 180

P1

x = 25

P1

70°

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 4 marks · Calculator

Algebra

Lesson 3 of 7

“

Amy is 3 times as old as Ben. In 8 years' time, Amy will be twice as old as Ben. How old is Amy now?

HOW TO ANSWER IT

Command word: Calculator. Worth 4 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Algebra

Lesson 3 of 7

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

Ben = b and Amy = 3b, or equivalent

P1

3b + 8 = 2(b + 8)

P1

b = 8

P1

Amy is 24

A1

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