M4.8.7 Chapter 7

Two equations on a graph

The lines y = x + 1 and y = 7 − x cross at (3, 4). What does that point tell me?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A The two lines have the same gradient.
You picked A. Read on. At the end, the chapter comes back to your guess.
B x = 3 and y = 4 make both equations true.
You picked B. Read on. At the end, the chapter comes back to your guess.
C x = 4 and y = 3 make both equations true.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Solve two equations by finding where their lines cross
Explain why parallel lines give no answer

The equation y = x + 1 has many pairs that fit it. x = 0 and y = 1 fit. So do x = 2 and y = 3. Each pair is a point, and the points form a straight line.

A second equation, y = 7 − x, makes a second line. A pair that fits both equations must be on both lines. Two different lines meet at one point at most.

Draw it first

Draw both lines

1 Draw the line y = x + 1. It crosses the y axis at 1 and rises 1 for each 1 across.
2 Draw the line y = 7 − x. It crosses the y axis at 7 and falls 1 for each 1 across.
3 Mark the point where they cross. Read across and up: x = 3 and y = 4.

The crossing point is the intersection of the lines. Its coordinates are the solution of the two equations: the pair that makes both true.

Worked example

Solve y = x + 1 and y = 7 − x by drawing graphs.

Table for y = x + 1. When x is 0, 3 and 6, y is 1, 4 and 7.
Table for y = 7 − x. When x is 0, 3 and 6, y is 7, 4 and 1.
Plot the points and draw both lines. They cross at (3, 4).
So x = 3 and y = 4.
Check in both equations: 3 + 1 = 4 and 7 − 3 = 4.

Answer: x = 3, y = 4

A graph shows a solution to the accuracy of the drawing. When lines cross between the grid lines, you can only estimate. Take y = 2x and y = 5 − x.

P is between 1 and 2 across, and between 3 and 4 up. The graph gives an estimate.

Algebra gives the exact answer. Put the right sides equal: 2x = 5 − x, so 3x = 5 and x = 5/3, which is about 1.7. Then y = 2x is about 3.3. A graph and algebra agree, but only algebra is exact.

Some lines never cross. y = 2x + 1 and y = 2x + 5 both go up 2 for each 1 across. They stay the same distance apart. Lines with the same gradient and different values of c are parallel.

The lines never cross, so no pair of numbers fits both equations.

Worked example

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

Both lines have gradient 2.
The lines cross the y axis at 1 and at 5. These are different, so the lines are parallel.
Check with algebra. Put the right sides equal: 2x + 1 = 2x + 5. Take 2x from both sides: 1 = 5.
That is false for every x, so there is no solution.

Answer: No solution. Both gradients are 2.

There is one more case. The equations y = 2x + 1 and 2y = 4x + 2 look different. Divide the second by 2 and you get y = 2x + 1 again. The two equations make the same line, so every point on it is a solution. There are infinitely many solutions.

Two equations for one line: infinitely many solutions.

Worked example

Two tanks. Tank A holds 40 litres and gains 5 litres each minute. Tank B holds 10 litres and gains 8 litres each minute. When do they hold the same?

After t minutes tank A holds V = 5t + 40 litres and tank B holds V = 8t + 10 litres.
They hold the same where the lines cross, so 5t + 40 = 8t + 10.
Take 5t and 10 from both sides: 30 = 3t, so t = 10.
Find V from tank A: V = 5 × 10 + 40 = 90 litres.
Check with tank B: V = 8 × 10 + 10 = 90 litres.

Answer: After 10 minutes, with 90 litres in each tank.

Tank B starts lower but fills faster, so the lines cross at P.

See it three ways

With things

Try x = 1, 2, 3 and 4 in both rules. The rule y = 2x gives 2, 4, 6 and 8. The rule y = 9 − x gives 8, 7, 6 and 5.

As a picture

In words

Both rules give 6 when x = 3. The two lines cross at the point where x = 3 and y = 6.

In symbols

2x = 9 − x, so 3x = 9 and x = 3. Then y = 2 × 3 = 6.

Try it

Find where two lines cross

You need: squared paper, a ruler, a pencil.

Make a table for y = 3x and for y = x + 4, with x from 0 to 4.
Plot the points and draw both lines with the ruler.
Read the point where the lines cross, then check it in both rules.

What you will see: The lines cross at x = 2, y = 6. Both rules give 6 when x = 2: 3 × 2 = 6 and 2 + 4 = 6.

If you cannot try it, this is what happens.

A wrong idea many people have

If two lines do not cross at a grid point, there is no solution.

Why it feels right: Whole numbers are quick to read, so a crossing between the grid lines feels like a miss.

The test that breaks it: y = 2x and y = 5 − x cross at x = 5/3, which is about 1.7. Put in x = 5/3: 2 × 5/3 = 10/3, and 5 − 5/3 = 10/3. Both sides are equal.

What is true: Lines with different gradients always cross, at one point. Its coordinates need not be whole numbers.

A wrong idea many people have

Any two lines meet somewhere.

Why it feels right: Lines are drawn as going on for ever, so they feel as if they must meet.

The test that breaks it: y = 2x + 1 and y = 2x + 5 are always 4 apart going up. At x = 0 they are 1 and 5, and at x = 10 they are 21 and 25.

What is true: Lines with the same gradient and different values of c never meet. Two equations like that have no solution.

Key words

intersection: the point where two lines cross.
solution: the pair of numbers that makes both equations true: the coordinates of the intersection.

Back to your guess

The answer is B: x = 3 and y = 4 make both equations true.

A point on a line fits that line's equation. A point on both lines fits both equations. So the crossing point (3, 4) means x = 3 and y = 4: 3 + 1 = 4 and 7 − 3 = 4.

Keep it

Copy this into your exercise book.

The crossing point of two lines is the solution. Parallel lines: no solution. The same line: infinitely many. A graph estimates, algebra is exact.

Draw: Two lines crossing at a point, with the point named and its x and y read off the axes.