Linear functions and graphs

Gradient, intercepts, the forms of a straight line, sketching lines, and solving simultaneous equations by graph and by algebra.

Gradient: the steepness of a line

The $m$ tells you how much $y$ changes when $x$ goes up by $1$. Positive gradients go up to the right, negative gradients go down, and horizontal lines have gradient $0$.

Gradient between two points

$$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$

Drag the two points

The line always passes through A and B. Watch how the rise, run and gradient change.

Try this

Make a line with gradient $-2$.
Make a horizontal line. What is its gradient?
Put A and B directly above each other. Why does the gradient become undefined?

Two forms of a straight line

Gradient–intercept form

$$y = mx + b$$

Read the gradient $m$ and the $y$-intercept $b$ straight off. Best for sketching from the $y$-axis.

General form

$$ax + by + c = 0$$

All terms on one side, usually with whole numbers and $a > 0$. Best for finding both intercepts quickly.

Example 1●

Find the gradient and $y$-intercept of $3x - 2y + 8 = 0$.

$-2y = -3x - 8$

Make y the subject: move 3x and 8 to the other side.

$y = \tfrac{3}{2}x + 4$

Divide every term by −2.

Gradient $m = \tfrac{3}{2}$, $y$-intercept $4$.

Sketching a line from its intercepts

To sketch $ax + by + c = 0$, find where it meets each axis. Put $x = 0$ to find the $y$-intercept, and $y = 0$ to find the $x$-intercept. Then join the two points.

Example 2●

Sketch $2x + 5y - 10 = 0$.

$x = 0$: $5y = 10$, so $y = 2$. The $y$-intercept is $(0, 2)$.

On the y-axis, x = 0.

$y = 0$: $2x = 10$, so $x = 5$. The $x$-intercept is $(5, 0)$.

On the x-axis, y = 0.

Draw the line through $(0, 2)$ and $(5, 0)$.

Two points fix a line. Label both intercepts on your sketch.

Simultaneous equations

Two lines usually meet at exactly one point. The coordinates of that point satisfy both equations, so finding the intersection solves the pair of equations.

Where do two lines meet?

Try this

Make the lines parallel. How many solutions are there now?
Make the two lines identical. What happens?

Example 3: elimination●●

Solve $3x + 2y = 7$ and $x - 2y = 5$.

Add: $4x = 12$, so $x = 3$.

The y terms are +2y and −2y, so adding the equations eliminates y.

$3 - 2y = 5$, so $-2y = 2$ and $y = -1$.

Substitute x = 3 into either equation.

Check: $3(3) + 2(-1) = 7$ ✓. Solution $x = 3$, $y = -1$, so the lines meet at $(3, -1)$.

Always check in the other equation.

Example 4: substitution●●

Solve $y = 2x - 1$ and $4x - y = 7$.

$4x - (2x - 1) = 7$

The first equation already gives y, so substitute it into the second.

$2x + 1 = 7$, so $x = 3$.

Careful with the minus sign in front of the bracket.

$y = 2(3) - 1 = 5$. The solution is $(3, 5)$.

Flashcards

Gradient formula

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

Gradient–intercept form

$y = mx + b$: gradient $m$, $y$-intercept $b$.

General form

$ax + by + c = 0$

Gradient of a horizontal line

$0$

Gradient of a vertical line

Undefined (the run is $0$).

Parallel lines

Same gradient. As simultaneous equations: no solution.

Practice

Find the gradient of the line through $(1, 4)$ and $(5, -2)$.
The line $4x + 2y - 6 = 0$ has gradient:
Two lines have the same gradient and different $y$-intercepts. How many solutions does the pair have?
Solve $x + y = 10$ and $x - y = -2$. Enter $x$.
The gradient of $y = 5 - 3x$ is:
Find the $x$-intercept of $3x - 4y - 12 = 0$.
Which line is horizontal?
Find the gradient of any line parallel to $2x + y = 7$.