Year 11 General Linear Equations And Their Graphs

Finding The Equation Of A Straight-Line Graph From Its Slope or a Point

10 practice questions 2 video lessons Theory + worked examples

Theory

Learn how to write the equation of a straight line when you are given its slope together with either the y-intercept, the x-intercept, or any other point on the line. Use the point-slope form $y - y_{1} = m (x - x_{1})$ and rearrange into the slope-intercept form $y = m x + c$ . Includes the special cases of horizontal and vertical lines.

The slope-intercept form $y = m x + c$ lets you write the equation of a line straight away — provided you know its slope $m$ and its $y$ -intercept $c$ . When the question hands you a slope together with some other point on the line, use the point-slope form instead:

y - y_{1} = m (x - x_{1})

Here $m$ is the slope and $(x_{1}, y_{1})$ is any known point on the line. Solve this for $y$ and you have the slope-intercept form.

The x-intercept is just a special point: the place where the line crosses the $x$ -axis, written $(a, 0)$ . A horizontal line has slope $0$ and equation $y = b$ . A vertical line has undefined slope and equation $x = a$ .

The first diagram shows a line drawn through a given point with a chosen slope — the situation the point-slope form is built for. The second shows the two special cases: a horizontal line (constant $y$ ) and a vertical line (constant $x$ ).

Slope

m = 2

through the point

(1, 3)

: use

y - 3 = 2 (x - 1)

, giving

y = 2 x + 1

Horizontal line

y = 2

(slope

0

) and vertical line

x = - 1

(slope undefined).

Pick the right form based on what the question gives you:

y - y_{1} = m (x - x_{1}) point-slope form

y - y_{1} = m (x - x_{1})

y = m x + c slope-intercept form

y = m x + c

Choosing the right approach

You are given…	Use…
slope $m$ and $y$ -intercept $c$	$y = m x + c$ directly
slope $m$ and a point $(x_{1}, y_{1})$	$y - y_{1} = m (x - x_{1})$
slope $m$ and $x$ -intercept $(a, 0)$	$y - 0 = m (x - a)$
horizontal line through $(a, b)$	$y = b$
vertical line through $(a, b)$	$x = a$

Tip. The point-slope form always works when you have a slope and a point — even if the point is the y-intercept or the x-intercept. The other forms are shortcuts for special cases.

How to find the equation from a slope and a point

Substitute the slope $m$ and the point $(x_{1}, y_{1})$ into $y - y_{1} = m (x - x_{1})$ .
Expand the brackets on the right-hand side.
Add $y_{1}$ to both sides to isolate $y$ .
Tidy into the form $y = m x + c$ .

EXAMPLE 1 — SLOPE AND Y-INTERCEPT

Find the equation of the line with slope

3

and

y

-intercept

(0, 5)

SOLUTION

Substitute $m = 3$ and $c = 5$ into $y = m x + c$ .

$y$	$=$	$m x + c$
$y$	$=$	$3 x + 5$

y = 3 x + 5

EXAMPLE 2 — SLOPE AND A POINT

Find the equation of the line with slope

7

passing through

(- 1, 2)

SOLUTION

Substitute $m = 7$ , $x_{1} = - 1$ , $y_{1} = 2$ into the point-slope form.

$y - y_{1}$	$=$	$m (x - x_{1})$
$y - 2$	$=$	$7 (x - (- 1))$
$y - 2$	$=$	$7 (x + 1)$
$y - 2$	$=$	$7 x + 7$
$y$	$=$	$7 x + 9$

y = 7 x + 9

EXAMPLE 3 — SLOPE AND X-INTERCEPT

Find the equation of the line with slope

- 2

and

x

-intercept at

(2, 0)

SOLUTION

The $x$ -intercept is the point $(2, 0)$ , so $y_{1} = 0$ . Apply the point-slope form.

$y - 0$	$=$	$- 2 (x - 2)$
$y$	$=$	$- 2 x + 4$

y = - 2 x + 4

EXAMPLE 4 — HORIZONTAL AND VERTICAL LINES

Find: (a) the horizontal line through

(3, 4)

, and (b) the vertical line through

(- 2, 5)

SOLUTION

(a) Horizontal lines have constant $y$ . The $y$ -coordinate is $4$ .

y

=

4

(b) Vertical lines have constant $x$ . The $x$ -coordinate is $- 2$ .

x

=

- 2

y = 4, x = - 2

Common pitfalls

Watch the minus signs. The point-slope form is

y - y_{1} = m (x - x_{1})

. If

(x_{1}, y_{1}) = (- 1, 2)

, the negative coordinate flips the minus to a plus:

x - (- 1) = x + 1

The $x$ -intercept is $(a, 0)$ , not $(0, a)$ . When a question gives the

x

-intercept as

(a, 0)

, the value

y_{1} = 0

, not

y_{1} = a

. Mixing this up changes the equation completely.

Don't forget to expand and simplify. Leaving the answer in the form

y - 2 = 7 (x + 1)

is technically correct but is not the slope-intercept form. Expand and add

y_{1}

to both sides to get

y = 7 x + 9

Horizontal vs vertical confusion. Horizontal lines have equation

y = constant

(the

y

-value never changes). Vertical lines have equation

x = constant

. Picture the line first, then write the equation.

Frequently asked questions

What is the point-slope form of a line?

The point-slope form is y minus y1 equals m times the quantity x minus x1, where m is the slope and (x1, y1) is any known point on the line. Solve for y to rearrange it into slope-intercept form y equals m x plus c.

When do I use point-slope form instead of slope-intercept form?

Use slope-intercept form y equals m x plus c when the question gives you the slope and the y-intercept directly. Use point-slope form when you have the slope and any other point on the line, including the x-intercept.

How do I find the equation of a line through a point with a given slope?

Substitute the slope m and the coordinates (x1, y1) of the point into y minus y1 equals m times (x minus x1). Expand the brackets, then add y1 to both sides to put it in the form y equals m x plus c.

What is the equation of a horizontal line through a point?

A horizontal line through the point (a, b) has equation y equals b. The slope is 0 and the y-value stays constant at b for every x.

What is the equation of a vertical line through a point?

A vertical line through the point (a, b) has equation x equals a. The slope is undefined and the x-value stays constant at a for every y.

What happens to the point-slope form when the point has negative coordinates?

The subtraction of a negative becomes addition. If (x1, y1) is (-1, 2), then y minus 2 equals m times the quantity x minus negative 1, which simplifies to y minus 2 equals m times (x plus 1).