Linear Equations in Two Variables

6.2

Linear Equations in Two Variables

Definition

A linear equation that has two variables and the power of each variable is \(1\).

General form: \(ax+by=c\), where \(a\) and \(b\) are non-zero.

Example

Determine whether \(\dfrac{m}{5}+7=12n\) is a linear equation in two variables.

\(\dfrac{m}{5}+7=12n\) is a linear equation in two variables.

This is because the equation has two variables, \(m\) and \(n\), and the power of each variable is \(1\).

Form linear equations in two variables:

Example

The difference between two numbers is \(27\).

Let the two numbers be \(r\) and \(s\) respectively.

The linear equation is

\(r-s=27\).

Determine the possible solutions of linear equations in two variables:

Has many possible pairs of values of solutions.

Steps:

Choose a value for one of the variables.
Substitute the value into the linear equation.
Solve the equation to find the value of the other variable.

Example

State three possible pairs of solutions for \(y=1-2x\).

When \(x=0\),

\(\begin{aligned} y&=1-2(0) \\\\&=1. \end{aligned}\)

When \(x=1\),

\(\begin{aligned} y&=1-2(1) \\\\&=-1. \end{aligned}\)

When \(x=2\),

\(\begin{aligned} y&=1-2(2) \\\\&=-3. \end{aligned}\)

Thus, three possible pairs of solutions are

\(x=0, y=1; x=1, y=-1,\) and \(x=2; y=-3.\)

Each pair of solutions can be writen in the ordered pair \((x,y)\).
Examples: \((0,1), (1,-1)\) and \((2, -3)\)

Represent graphically linear equations in two variables:

Example