Developing Linear Formulas
There are different formulas we can use to generate linear equations. Which one we choose depends on the information we are given.
- Gradient-intercept form: $y=mx+b$y=mx+b, where $m$m is the gradient and $b$b is the $y$y-intercept. We use this form when we know/ can calculate the gradient and $y$y-intercept.
- Gradient-point form: $y-y_1=m\left(x-x_1\right)$y−y1=m(x−x1), where $m$m is the gradient and $\left(x_1,y_1\right)$(x1,y1) is a point on the line. We use this form when we know/ can calculate the gradient and a point on the line.
- Two point form: $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$y−y1x−x1=y2−y1x2−x1, where $\left(x_1,y_1\right)$(x1,y1) and $\left(x_2,y_2\right)$(x2,y2) are points on the line. We use this form when we know two points on the line.
Worked Examples
Question 1
Write down the equation of a line whose gradient is $2$2 and crosses the $y$y-axis at $\left(0,1\right)$(0,1).
Express your answer in gradient-intercept form.
Question 2
The table shows the linear relationship between the number of plastic chairs manufactured and the total manufacturing cost.
| Number of plastic chairs | $2$2 | $4$4 | $7$7 |
|---|---|---|---|
| Cost (dollars) | $135$135 | $185$185 | $260$260 |
What is the slope of the function?
Write an equation to represent the total manufacturing cost, $y$y, as a function of the number of plastic chairs manufactured, $x$x.
What is the $y$y-intercept?
What does this $y$y-intercept tell you?
The fixed cost of manufacturing
AThe variable cost of manufacturing
BThe sale price of each plastic chair
CThe profit generated from selling each plastic chair
DWhat does the slope of the function represent?
The fixed cost of manufacturing
AThe cost of producing $0$0 plastic chairs
BThe sale price of each plastic chair
CThe variable cost of manufacturing
DFind the total cost of manufacturing $13$13 plastic chairs.
Question 3
A line passes through the point $A$A$\left(-4,5\right)$(−4,5) and has a gradient of $3\frac{1}{2}$312. Using the point-gradient formula, express the equation of the line in gradient intercept form .