Using Graphs to Solve Linear Equations
Determine the Number of Solutions of a Linear System
There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.
We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in Example 5.2 through Example 5.6 all had two intersecting lines. Each system had one solution.
A system with parallel lines, like Example 5.7, has no solution. What happened in Example 5.8? The equations have coincident lines, and so the system had infinitely many solutions.
We'll organize these results in Figure 5.3 below:
|
Graph |
Number of solutions |
|
2 intersecting lines |
1 |
|
Parallel lines |
None |
|
Same line |
Infinitely many |
Figure 5.3
Parallel lines have the same slope but different \(y\) -intercepts. So, if we write both equations in a system of linear equations in slope-intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in Example 5.7.
The first line is in slope–intercept form.
|
\(\left\{\begin{aligned} y &=\frac{1}{2} x-3 \\ x-2 y &=4 \end{aligned}\right.\) |
|
|
The first line is in slope-intercept form. \( \begin{array}{c} y=\frac{1}{2} x-3 \\ m=\frac{1}{2}, b=-3 \end{array} \) |
If we solve the second equation for \(y\), we get \( \begin{aligned} x-2 y &=4 \\ -2 y &=-x+4 \\ y &=\frac{1}{2} x-2 \\ m=\frac{1}{2}, b &=-2 |
The two lines have the same slope but different y-intercepts. They are parallel lines.
Figure 5.4 shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.
|
Number of Solutions of a Linear System of Equations |
|||
|
Slopes |
Intercepts |
Type of Lines |
Number of Solutions |
|
Different |
|
Intersecting |
1 point |
|
Same |
Different |
Parallel |
No solution |
|
Same |
Same |
Coincident |
Infinitely many solutions |
Figure 5.4
Let's take one more look at our equations in Example 5.7 that gave us parallel lines.
\(\left\{\begin{array}{l}y=\frac{1}{2} x-3 \\ x-2 y=4\end{array}\right.\)
When both lines were in slope-intercept form we had:
\(y=\frac{1}{2} x-3 \qquad \qquad y=\frac{1}{2} x-2\)
Do you recognize that it is impossible to have a single ordered pair \((x,y)\) that is a solution to both of those equations?
We call a system of equations like this an inconsistent system. It has no solution.
A system of equations that has at least one solution is called a consistent system.
Consistent and Inconsistent Systems
A consistent system of equations is a system of equations with at least one solution.
An inconsistent system of equations is a system of equations with no solution.
We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent equations, they each have their own set of solutions. Intersecting lines and parallel lines are independent.
If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.
Independent and Dependent Equations
Two equations are independent if they have different solutions.
Two equations are dependent if all the solutions of one equation are also solutions of the other equation.
Let's sum this up by looking at the graphs of the three types of systems. See Figure 5.5 and Figure 5.6.
Figure 5.5
|
Lines |
Intersecting |
Parallel |
Coincident |
|
Number of solutions |
1 point |
No solution |
Infinitely many |
|
Consistent/inconsistent |
Consistent |
Inconsistent |
Consistent |
|
Dependent/independent |
Independent |
Independent |
Dependent |
Figure 5.6
Example 5.9
Without graphing, determine the number of solutions and then classify the system of equations: \(\left\{\begin{array}{l}y=3 x-1 \\ 6 x-2 y=12\end{array}\right.\)
Solution
| We will compare the slopes and intercepts of the two lines. | \(\left\{\begin{array}{l}y=3 x-1 \\ 6 x-2 y=12\end{array}\right.\). |
| The first equation is already in slope-intercept form. | \(y=3 x-1\) |
| Write the second equation in slope-intercept form. | \(\begin{aligned} 6 x-2 y &=12 \\-2 y &=-6 x+12 \\ \frac{-2 y}{-2} &=\frac{-6 x+12}{-2} \\ y &=3 x-6 \end{aligned}\) |
| Find the slope and intercept of each line. | \(\begin{array}{rlrl}y & =3 x-1 & \qquad y & =3 x-6 \\ m & =3 & m & =3 \\ b & =-1 & b & =-6\end{array}\) |
| Since the slopes are the same and \(y\)-intercepts are different, the lines are parallel. |
A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.
Try It 5.17
Without graphing, determine the number of solutions and then classify the system of equations.
\(\left\{\begin{array}{l}y=-2 x-4 \\ 4 x+2 y=9\end{array}\right.\)
Try It 5.18
Without graphing, determine the number of solutions and then classify the system of equations.
\(\left\{\begin{array}{l}y=\frac{1}{3} x-5 \\ x-3 y=6\end{array}\right.\)
Example 5.10
Without graphing, determine the number of solutions and then classify the system of equations: \(\left\{\begin{array}{l}2 x+y=-3 \\ x-5 y=5\end{array}\right.\).
Solution
| We will compare the slope and intercepts of the two lines. | \(\left\{\begin{array}{l}2 x+y=-3 \\ x-5 y=5\end{array}\right.\) | |
| Write both equations in slope-intercept form. | \(\begin{aligned} 2 x+y &=-3 \\ y &=-2 x-3 \end{aligned}\) | \(\begin{aligned} x-5 y=5 &=5 \\-5 y &=-x+5 \\ \frac{-5 y}{-5} &=\frac{-x+5}{-5} \\ y &=\frac{1}{5} x-1 \end{aligned}\) |
| Find the slope and intercept of each line. | \(\begin{aligned} y &=-2 x-3 \\ m &=-2 \\ b &=-3 \end{aligned}\) | \(\begin{aligned} y &=\frac{1}{5} x-1 \\ m &=\frac{1}{5} \\ b &=-1 \end{aligned}\) |
| Since the slopes are different, the lines intersect. | ||
A system of equations whose graphs are intersect has 1 solution and is consistent and independent.
Try It 5.19
Without graphing, determine the number of solutions and then classify the system of equations.
\(\left\{\begin{array}{l}3 x+2 y=2 \\ 2 x+y=1\end{array}\right.\)
Try It 5.20
Without graphing, determine the number of solutions and then classify the system of equations.
\(\left\{\begin{array}{l}x+4 y=12 \\ -x+y=3\end{array}\right.\)
Example 5.11
Without graphing, determine the number of solutions and then classify the system of equations.
\(\left\{\begin{array}{l}3 x-2 y=4 \\ y=\frac{3}{2} x-2\end{array}\right.\)
Solution
| We will compare the slopes and intercepts of the two lines. | \(\left\{\begin{array}{l}3 x-2 y=4 \\ y=\frac{3}{2} x-2\end{array}\right.\) |
| Write the first equation in slope-intercept form. | \(\begin{aligned} 3 x-2 y &=4 \\-2 y &=-3 x+4 \\ \frac{-2 y}{-2} &=\frac{-3 x+4}{-2} \\ y &=\frac{3}{2} x-2 \end{aligned}\) |
| The second equation is already in slope-intercept form. | \(y=\frac{3}{2} x-2\) |
| Since the slopes are the same, they have the same slope and same \(y\)-intercept and so the lines are coincident. |
A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.
Try It 5.21
Without graphing, determine the number of solutions and then classify the system of equations.
\(\left\{\begin{array}{l}4 x-5 y=20 \\ y=\frac{4}{5} x-4\end{array}\right.\)
Try It 5.22
Without graphing, determine the number of solutions and then classify the system of equations.
\(\left\{\begin{array}{l}-2 x-4 y=8 \\ y=-\frac{1}{2} x-2\end{array}\right.\)