Solving simultaneous equations graphically.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving simultaneous equations graphically

Equations and graphs

Lesson 1 of 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Equations and graphs

Lesson 1 of 6

Answer each one, then check.

1

What is the gradient of y = 3x + 2?

2

Where does y = 3x + 2 cross the y-axis?

3

Find y when x = 2 in y = 2x − 1.

4

What does "simultaneous" mean?

5

Rearrange x + y = 5 to make y the subject.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Equations and graphs

Lesson 1 of 6

1

What is the gradient of y = 3x + 2?

3

2

Where does y = 3x + 2 cross the y-axis?

(0, 2)

3

Find y when x = 2 in y = 2x − 1.

3

4

What does "simultaneous" mean?

Both equations true at the same time

5

Rearrange x + y = 5 to make y the subject.

y = 5 − x

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Equations and graphs

Lesson 1 of 6

1

Draw straight lines from equations.

2

Find where two lines intersect.

3

Read the solution to a pair of simultaneous equations.

4

Check the solution by substituting.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and graphs

Lesson 1 of 6

GRAPHICAL SOLUTION

The coordinates of the point where two lines cross satisfy both equations at once.

The solution to y = 2x − 1 and x + y = 5 is the point (2, 3), so x = 2 and y = 3.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Where the Lines Cross

Equations and graphs

Lesson 1 of 6

The intersection gives both x and y.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving Graphically

Equations and graphs

Lesson 1 of 6

Draw both lines carefully.

1

Rearrange

Make each equation y = mx + c if you need to

2

Make a table or use the gradient and intercept

Find at least three points per line

3

Draw both lines

Use a ruler and extend across the grid

4

Read the crossing point

Write down x and y

5

Check

Substitute into both equations

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solve by Drawing

Equations and graphs

Lesson 1 of 6

Solve y = 2x − 1 and x + y = 5 graphically.

1

Line 1 points

(0, −1), (2, 3), (4, 7)

2

Line 2: y = 5 − x

(0, 5), (2, 3), (5, 0)

3

Lines cross at

(2, 3)

4

Check line 1

2 × 2 − 1 = 3, correct

5

Check line 2

2 + 3 = 5, correct

ANSWER

x = 2 and y = 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

No Solutions

Equations and graphs

Lesson 1 of 6

Explain why y = 2x + 1 and y = 2x + 5 have no solutions.

1

Gradients

Both are 2

2

Parallel lines

They never meet

ANSWER

The lines are parallel, so there is no point that satisfies both equations.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Tips

Equations and graphs

Lesson 1 of 6

Get an accurate answer.

Ruler

Use a sharp pencil and a ruler.

Check

Always substitute back to check.

Rearranging

Get both equations into y = mx + c form before drawing.

Estimates

A graph gives an approximate answer unless the point is on a grid crossing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Equations and graphs

Lesson 1 of 6

Simultaneous equations

Equations that are both true for the same values.

Intersection

The point where two lines cross.

Gradient

How steep a line is.

Intercept

Where a line crosses an axis.

Parallel

Lines with the same gradient.

Solution

The values of x and y that satisfy both equations.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and graphs

Lesson 1 of 6

YOUR TASK

Find the Crossing Point

12 minutes

Draw y = x − 1 and y = 5 − x on the same axes for x from 0 to 5. Read the solution.

1

Make a table of values for each line.

2

Read where they cross.

WHAT A GOOD ANSWER SHOWS

The lines cross at (3, 2), so x = 3 and y = 2.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Equations and graphs

Lesson 1 of 6

Draw a line from its equation.

Rearrange to y = mx + c.

Draw two lines on one grid.

Read the intersection.

Write x and y values.

Check by substitution.

Explain why parallel lines have no solution.

Draw accurately.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Solution = coordinates of the intersection.

Check in both equations.

Parallel lines have no solution.

Rearrange first to help you draw.

EXAM FOCUS

On the grid, draw the graph of x + y = 5. Hence solve y = 2x − 1 and x + y = 5. (4 marks)

Plot at least three points for each line. Circle the intersection.

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