When given the slope and y-axis intercept of a line, it is very easy to determine its equation, using the slope-intercept form of a line y=a+bx, since b is the slope and a is the y-intercept. Simply substitute the two values into the equation.

Examples

Example 1

Write down the equation of a line whose slope is -8 and crosses the y-axis at -9. Express your answer in slope-intercept (gradient-intercept) form.

Worked Solution

Create a strategy

Use the slope-intercept form and substitute the slope for b and the value of the y-intercept for a.

Apply the idea

From the information we have b=-8 and a=-9. So the equation is: y=-9 -8x

Idea summary

To find the equation of a line given the slope and y-axis intercept, use the slope-intercept form of a line:

\displaystyle y=a+bx

\bm{b}

is the slope

\bm{a}

is the y-intercept

Given the slope and one known point on the line

When given the slope of the line and one point that lies on that line, the equation of that line can be found by following a few simple steps.

Substitute the slope into the slope-intercept form of the equation y = a +bx.
Using the coordinate point given, substitute each x and y-value into the equation and solve for the remaining unknown a, the y-intercept..
State the equation for the line

This process can be streamlined using the point-slope formula y - y_1 = b(x - x_1) where b is the slope and (x_1,\, y_1) is the known coordinate on the line.

If using this formula, the brackets need to be expanded and y needs to be made the subject of the equation, so that the equation is displayed in its simplest slope-intercept form.

Examples

Example 2

Given that the line y=a + bx has a slope of -2 and passes through \left(-7, 5\right).

Find a, the value of the y-intercept of the line.

Worked Solution

Create a strategy

Use the slope-intercept form y=a + bx and substitute the given values.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle a + bx	Use the slope-intercept form
\displaystyle a + bx	\displaystyle =	\displaystyle y	Swap the sides
\displaystyle a + (-2)\times(-7)	\displaystyle =	\displaystyle 5	Substitute b=2, x=-7, and y=5
\displaystyle a + 14	\displaystyle =	\displaystyle 5	Simplify
\displaystyle a	\displaystyle =	\displaystyle -9	Subtract 14 from both sides

Find the equation of the line in the form y=a+bx.

Worked Solution

Create a strategy

Substitute the slope and the value of a found from part (a) into y=a + bx.

Apply the idea

y=-2x-9

Reflect and check

We can also use the point-slope formula: y-y_1=b\left(x-x_1\right) to find the equation of the line.

\displaystyle y-y_1	\displaystyle =	\displaystyle b\left(x-x_1\right)	Use the point-slope formula
\displaystyle y-5	\displaystyle =	\displaystyle -2\left(x-(-7)\right)	Susbtitute \left(-7, 5\right) and b=-2
\displaystyle y-5	\displaystyle =	\displaystyle -2\left(x+7\right)	Evaluate the adjacent signs
\displaystyle y-5	\displaystyle =	\displaystyle -2x-14	Expand the brackets
\displaystyle y-5+5	\displaystyle =	\displaystyle -2x-14+5	Add 5 to both sides
\displaystyle y	\displaystyle =	\displaystyle -2x-9	Evaluate

Idea summary

When given the slope of the line and one point that lies on that line, use the following steps.

Substitute the slope into the slope-intercept form of the equation y = a +bx.
Using the coordinate point given, substitute each x and y-value into the equation and solve for the remaining unknown a, the y-intercept..
State the equation for the line

We can also find the equation of a line given the slope and a point by using the point-slope formula:

\displaystyle y-y_1=b\left(x-x_1\right)

\bm{b}

is the slope

\bm{(x_1,y_1)}

are the coordinates of the given point

Given two known coordinate points on the line

When only provided with two coordinate points from the graph, the equation can be found by using the following steps:

Let the given coordinates equal \left(x_1, y_1\right) and \left(x_2, y_2\right) respectively.
Use the coordinates of the two points to determine the slope b=\dfrac{(y_2-y_1)}{(x_2-x_1)}.
Substitute this value for the slope into the equation. There is now only one unknown, a.
Substitute the coordinates of one of the two points on the line into this new equation and solve for the unknown a, the y-intercept.
Substitute the values of b and a into the general equation y = a +bx to obtain the equation of the straight line.

Examples

Example 3

A line passes through the points \left(3, -5\right) and \left(-7, 2\right).

Find the slope (gradient) of the line.

Worked Solution

Create a strategy

Substitute the given two points into the slope formula.

Apply the idea

\displaystyle b	\displaystyle =	\displaystyle \dfrac{y_2-y_1}{x_2-x_1}	Use the slope formula
	\displaystyle =	\displaystyle \dfrac{2-(-5)}{-7-3}	Substitute the coordinates
	\displaystyle =	\displaystyle -\dfrac{7}{10}	Evaluate

Find the equation of the line by substituting the slope and one point into y-y_1=b\left(x-x_1\right).

Worked Solution

Create a strategy

Use the point-slope formula.

Apply the idea

Let \left(-7, 2\right) be our one point and from part (a) we have b=-\dfrac{7}{10}.

\displaystyle y-y_1	\displaystyle =	\displaystyle b\left(x-x_1\right)	Write the point-slope formula
\displaystyle y-2	\displaystyle =	\displaystyle -\dfrac{7}{10}\left(x-(-7)\right)	Susbtitute \left(-7, 2\right) and b=-\dfrac{7}{10}
\displaystyle y-2	\displaystyle =	\displaystyle -\dfrac{7}{10}\left(x+7\right)	Evaluate the adjacent signs
\displaystyle y-2	\displaystyle =	\displaystyle -\dfrac{7}{10}x-\dfrac{49}{10}	Expand the brackets
\displaystyle y-2+2	\displaystyle =	\displaystyle -\dfrac{7}{10}x-\dfrac{49}{10}+2	Add 2 to both sides
\displaystyle y	\displaystyle =	\displaystyle -\dfrac{7}{10}x-\dfrac{29}{10}	Evaluate

Idea summary

To find the equation of a line given two coordinate points on the line, we can use the following steps:

Let the given coordinates equal \left(x_1, y_1\right) and \left(x_2, y_2\right) respectively.
Use the coordinates of the two points to determine the slope b=\dfrac{(y_2-y_1)}{(x_2-x_1)}.
Substitute the coordinates of one of the two points on the line and the value of b into the point-slope formula y-y_1=b\left(x-x_1\right) to obtain the equation of the straight line.

Outcomes

U1.AoS4.1

the properties of linear functions and their graphs

U1.AoS4.3

the forms, rules, graphical images and tables for linear relations and equations

3.03 Equation of a straight line

Introduction

Ideas

Given the slope and y-axis intercept

Examples

Example 1

Given the slope and one known point on the line

Examples

Example 2

Given two known coordinate points on the line

Examples

Example 3

Outcomes

U1.AoS4.1

U1.AoS4.3

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