Learning Objectives
- Define a linear equation
- Determine if an equation in two variables is linear
- Write a linear equation in standard form
- Write a linear equation in slope-intercept form
Key words
- Linear equation: an equation where each term has only one variable and the highest exponent of each variable is 1
- Standard form: [latex]ax+by=c[/latex] where [latex]a,\,b,\,c[/latex] are constants
Definition of Linear Equations
A linear equation is an equation with the combination of a constant and one or more variable terms. Each variable term contains only one variable, and the exponent of the variable is one.
ExAMPLES
- [latex]2x+3=5x-4[/latex] is a linear equation in one variable
- [latex]2x+3y=5[/latex] is a linear equation in two variables
- [latex]5x-3y+2z=12[/latex] is a linear equation in three variables
- [latex]a_{1}x_{1}+a_{2}x_{2} + ... + a_{n}x_{n} = c[/latex] is the general form of a linear equation
Remember that if a variable is in the denominator with an exponent of 1, it really has an exponent of –1. For example, [latex]\frac{1}{x}=x^{-1}[/latex]. Therefore, for a variable to have an exponent of 1, it is always in the numerator.
Linear Equations in Two Variables
A linear equation in two variables is a linear equation with two variables where the exponents on the variables are 1 and each term contains only one variable.
ExAMPLE
The following are examples of two-variable linear equations:
- [latex]3x + 5y = 7[/latex]
- [latex]y = \frac{2}{3}x - 3[/latex]
- [latex]\left(y-2\right)=\frac{-2}{7}\left(x+3\right)[/latex]
The following are NOT two-variable linear equations:
- [latex]x^2 + y^2 = 9[/latex]
- This equation is not linear because the exponent of the [latex]x[/latex] and [latex]y[/latex]-variables is not 1.
- [latex]\sqrt{2x-3}=y[/latex]
- This equation is not linear because the exponent on the [latex]x[/latex]-variable is not 1.
- [latex]\frac{2}{x}=3y[/latex]
- This equation is not linear because the exponent on the [latex]x[/latex]-variable is not 1. (it is -1)
The standard form of a linear equation in two variable is: [latex]ax+by=c[/latex] where [latex]a,\,b,\,c[/latex] are constants.
For example the equation [latex]2x+y = 5[/latex] is a two-variable linear equation written in standard form.
Linear equations are not always given in standard form but can always be simplified to standard form using the addition and multiplication properties of equality.
Example
Determine if the equation is linear and is in standard form. If it is not, rewrite it in standard form.
1. [latex]3x-4y=9[/latex]
2. [latex]y=x+6[/latex]
3. [latex]4x+6y-9=0[/latex]
4. [latex]3xy=8[/latex]
5. [latex]3(x-2)+5y=12[/latex]
Solution
1. [latex]3x-4y=9[/latex] is in the form [latex]ax+by=c[/latex] so is in standard form.
2. [latex]y=x+6[/latex] is a linear equation but is not in standard form. To write it in standard form, all the variables must be on the same side, so we will subtract [latex]x[/latex] from both sides of the equation: [latex]\begin{equation}\begin{aligned}y=x+6 \\ y-x=6\end{aligned}\end{equation}[/latex]
3. [latex]4x+6y-9=0[/latex] is a linear equation but is not in standard form. We need to add [latex]9[/latex] to both sides of the equation: [latex]\begin{equation}\begin{aligned}4x+6y-9=0 \\ 4x+6y=9\end{aligned}\end{equation}[/latex]
4. [latex]3xy=8[/latex] is not a linear equation. Although each exponent on the variables is 1, there is more than one variable in the term [latex]3xy[/latex].
5. [latex]3(x-2)+5y=12[/latex] is a linear equation but is not in standard form. We need to distribute and add [latex]6[/latex] to both sides:
[latex]\begin{equation}\begin{aligned}3(x-2)+5y=12 \\ 3x-6+5y=12 \\ 3x+5y=18\end{aligned}\end{equation}[/latex]
Try It
Write the linear equation in standard form.
1. [latex]y=-3x+6[/latex]
2. [latex]2x+y-9=0[/latex]
3. [latex]3(2x-1)-7y=4[/latex]
Another way to write a linear equation in two variables is called slope-intercept form. Slope-Intercept form: [latex]y=mx+b[/latex] where [latex]m[/latex] and [latex]b[/latex] are constants. We will find out later exactly what these constants represent. For example [latex]y=3x-6[/latex] and [latex]y=-\frac{2}{3}x+\frac{4}{5}[/latex] are linear equations in slope-intercept form.
If the linear equation is not given in slope-intercept form we can always rearrange it using the addition and multiplication properties of equality.
Examples
Write the linear equations in slope-intercept form.
1. [latex]2x+y=7[/latex]
2. [latex]x-3y=6[/latex]
3. [latex]2x-5y=4[/latex]
4. [latex]2(x-3)-(2y-1)=1[/latex]
Solution
1.
[latex]\begin{equation}\begin{aligned}2x+y & =7 \;\;\;\;\;\;\;\;\;\;\text{Subtract }2x\text{ from both sides} \\ y & =-2x+7\end{aligned}\end{equation}[/latex]
2.
[latex]\begin{equation}\begin{aligned}x-3y & =6 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Subtract }\;x\text { from both sides}\\ -3y & =-x+6 \;\;\;\;\;\;\;\;\;\;\text{Divide both sides by }-3 \\ y & =\frac{-x+6}{-3} \;\;\;\;\;\;\;\;\;\text{Distribute the }-3 \\ y & =\frac{1}{3}x-2\end{aligned}\end{equation}[/latex]
3.
[latex]\begin{equation}\begin{aligned}2x-5y & =4 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Subtract } \;2x \\-5y & =-2x+4\;\;\;\;\;\;\;\;\;\;\text{Divide by }-5 \\y & = \frac{-2x+4}{-5}\;\;\;\;\;\;\;\;\;\text{Distribute} \\ y & =\frac{2}{5}x-\frac{4}{5}\end{aligned}\end{equation}[/latex]
4.
[latex]\begin{equation}\begin{aligned}2(x-3)-(2y-1) & =1\;\;\;\;\;\;\;\;\;\;\text{Distribute} \\2x-6-2y+1 & =1 \;\;\;\;\;\;\;\;\;\;\text{Simplify} \\ 2x-2y-5 & =1 \;\;\;\;\;\;\;\;\;\;\text{Add } 5 \\ 2x-2y & =6 \;\;\;\;\;\;\;\;\;\;\text{Subtract}\;2x \\-2y & = -2x+6 \;\;\;\;\;\;\;\;\;\;\text{Divide by}\; -2 \\ y & = \frac{-2x+6}{-2} \;\;\;\;\;\;\;\;\;\;\text{Distribute} \\ y & =x-3 \end{aligned}\end{equation}[/latex]
Try It
Write the linear equations in slope-intercept form.
1. [latex]x+4y=8[/latex]
2. [latex]3x+2y=4[/latex]
4. [latex]2(x+4)-(y-3)=2[/latex]
