Writing Linear Equations

Concept

Write linear equations involving variables. This equation will be a linear combination of the two variables, and a constant can be present. When any linear equation is plotted on a graph, it will necessarily produce a straight line – hence the name: Linear equations.

Rules

1. Find the slope using the formula $\fn_phv m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
2. Use the slope for m and the coordinates of either point (as $\fn_phv x_{1}$ and $\fn_phv y_{1}$ ) in the formula $\fn_phv y-y_{1}=m(x-x_{1})$
3. Simplify the equation and solve for $\fn_phv y$ .

Example

It costs $5 to mail a package weighing 2 kg. To mail a package weighing 3 kg, the shipping cost is $6.50.
Write the linear equation that models the relationship between the weight of a package (x) and the shipping cost (y).

Solution

The data points are (2 kg, $5) and (3 kg, $6.50).
To find the slope, use the formula
$\fn_phv m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$\fn_phv m=\frac{6.5-5}{3-2}=\frac{1.5}{1}$
Then, Use the slope for m and the coordinates of either point (as $\fn_phv x_{1}$ and $\fn_phv y_{1}$ ) in the formula $\fn_phv y-y_{1}=m(x-x_{1})$
$\fn_phv y-5=1.5(x-2)$
Simplify the equation.
$\fn_phv y-5=1.5x-3$
Add 5 to both sides, and you get:
$\fn_phv {\color{Red} y=1.5x+2}$

Practice Writing Linear Equations

Practice Problem 1

Write the equation of a line with a slope of − $\fn_phv {\color{Teal} \frac{2}{5}}$ that goes through the point (5,−1).

Practice Problem 2

Write the equation of the line that passes through A (2,5) and B (-2,3).

Practice Problem 3

It costs $32 for 8 liters of gas. It costs $46 for 12 liters of gas.
Write the linear equation that models the relationship between the number of liters (x) and the cost of gas (y).

Algebra – the branch of mathematics that involves expressions with variables.

Variable – a symbol that represents an unknown quantity.

Equation – In algebra an equation can be defined as a mathematical statement consisting of an equal symbol between two algebraic expressions that have the same value.

A linear equation is an equation that is written for two different variables.

Pre-requisite Skills
Constant rate of Change
Slope
Direct Variation Equations (y = mx)
Slope-intercept Form of an Equation
Graphing Lines Using Intercepts