Chapter 2 Matrices
2.1 Systems of Equations
Learning Objectives
By the end of this section, you will be able to:
- Solve linear systems using graphing
- Solve linear systems using substitution
- Solve linear systems using elimination (addition)
- Classify systems as independent, inconsistent, or dependent
- Solve linear systems in 3 variables
Back when studying linear equations, we found the intersection of two lines. Doing so allowed us to solve interesting problems by finding a pair of values that satisfied two different equations. While we didn’t call it this at the time, we were solving a system of equations.
A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.
A solution to a system is a set of numerical values for each variable in the system that will satisfy all equations in the system at the same time.
Not every system will have exactly one solution, but we’ll look more closely at that later.
To check to see if an ordered pair is a solution to a system of equations, you would:
- Substitute the ordered pair into each equation in the system.
- Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.
Example 1
Determine whether the ordered pair [latex](5, 1)[/latex] is a solution to the given system of equations.
[latex]\begin{cases} x+3y=8\\2x-9=y\end{cases}[/latex]
Substitute the ordered pair [latex](5, 1)[/latex] into both equations.
[latex]\begin{array}{rclr} (5)+3(1) & = & 8 \\ 8 & = & 8& \text{True}\\ \\2(5)-9 &=&(1)\\1&=&1&\text{True}\end{array}[/latex]
The ordered pair [latex](5, 1)[/latex] satisfies both equations, so it is the solution to the system.
Solving a System by Graphing
There are three common methods for solving systems of linear equations with two variables. The first is solving by graphing.
To solve a system using the graphing method, you would:
- Graph both equations.
- The solution to the system is the intersection of the lines.
Example 2
Solve the following system of equations by graphing.
[latex]\begin {cases} 2x+y=-8\\x-y=-1\end {cases}[/latex]
Solve the first equation for [latex]y.[/latex]
[latex]\begin{array}{rcl}2x+y&=&-8\\y&=&-2x-8&\end{array}[/latex]
Solve the second equation for [latex]y.[/latex]
[latex]\begin{array}{rcl}x-y&=&-1\\y&=&x+1\end{array}[/latex]
Graph both equations on the same set of axes.
The lines appear to intersect at the point [latex](-3, -2).[/latex]
We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations.
[latex]\begin{array}{rclr}2(-3)+(-2) & = & -8 \\ -8 & = & -8 &\text{True}\\ \\(-3)-(-2) & =&-1\\-1&=&-1&\text{True}\end{array}[/latex]
The solution to the system is the ordered pair [latex](-3, -2)[/latex].
Exercise 1
While this method can work well enough when the solution values are both integers, it is not very useful when the intersection is no