Finding the equation of a straight line
There are three common methods to finding the equation of a straight line, based off of given information.
- Given the slope and $y$y-axis intercept.
- Given the slope and one known point on the line.
- Given two (or more) known points on the line.
Given the slope and $y$y-axis intercept
When given the slope and $y$y-axis intercept of a line, it is very easy to determine its equation, using the slope-intercept form of a line $y=a+bx$y=a+bx , since $b$b is the slope and $a$a is the $y$y-intercept. Simply substitute the two values into the equation.
Practice questions
Question 1
Write down the equation of a line whose slope is $-6$−6 and crosses the $y$y-axis at $\left(0,-2\right)$(0,−2).
Express your answer in slope-intercept form.
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 $y=a+bx$y=a+bx.
- Using the coordinate point given, substitute each $x$x and $y$y-value into the equation and solve for the remaining unknown $a$a, the $y$y-intercept.
- State the equation for the line.
This process can be streamlined using the point-slope formula
$y-y_1=b\left(x-x_1\right)$y−y1=b(x−x1),
where $b$b is the slope and $x_1,y_1$x1,y1 is the known coordinate on the line.
If using this formula, the brackets need to be expanded and $y$y needs to be made the subject of the equation, so that the equation is displayed in its simplest slope-intercept form.
Practice question
Question 2
Given that the line $y=a+bx$y=a+bx has a slope of $2$2 and passes through $\left(-9,4\right)$(−9,4):
Find $a$a, the value of the $y$y-intercept of the line.
Find the equation of the line in the form $y=a+bx$y=a+bx.
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 $\left(x_1,y_1\right)$(x1,y1) and $\left(x_2,y_2\right)$(x2,y2) respectively.
- Use the coordinates of the two points to determine the slope $b=\frac{y_2-y_1}{x_2-x_1}$b=y2−y1x2−x1.
- Substitute this value for the slope into the equation. There is now only one unknown, $a$a.
- Substitute the coordinates of one of the two points on the line into this new equation and solve for the unknown $a$a, the $y$y-intercept.
- Substitute the values of $b$b and $a$a into the general equation $y=a+bx$y=a+bx to obtain the equation of the straight line.
Practice question
Question 3
A line passes through the points $\left(-5,4\right)$(−5,4) and $\left(-8,3\right)$(−8,3).
Find the slope of the line.
Find the equation of the line by substituting the slope and one point into $y-y_1=b\left(x-x_1\right).$y−y1=b(x−x1).