Just like when we were solving single variable equations, when we solve a system of equations, there are different types of solutions that can exist: one solution, no solution, and infinitely many solutions. The solution to a system of equations is always the point(s) of intersection on the graph. If the two lines never intersect, then the system cannot have a solution. Let's review what it means for a system to have each of the types of solutions:
One solution: there is only one ordered pair, when substituted in the system for \(x\) and \(y\), that will create a true statement for each equation
No Solution: there is NO ordered pair that when substituted in to the system that will create a true statement for BOTH equations
Infinitely Many Solutions: there are an INFINITE amount of ordered pairs that can be substituted in to the system that will create true statements for BOTH equations
If you were to graph a system with one solution, the graphs of the two lines would intersect at a one single point. The graphs of the lines of system with no solution would never intersect with each other, meaning that the two lines would be parallel to each other. If the system has infinitely many solutions, then the graphs of the two lines would fall on top of one another, intersecting at all points along the two lines. Another way to think of this is by thinking about snails. Yep, snails. A snail went across the sidewalk in a straight line and made a trail. Another snail went across the same sidewalk in the exact same place as the first snail and also made a trail. At how many locations are the snails trails the same?
One solution: there is only one ordered pair, when substituted in the system for \(x\) and \(y\), that will create a true statement for each equation
No Solution: there is NO ordered pair that when substituted in to the system that will create a true statement for BOTH equations
Infinitely Many Solutions: there are an INFINITE amount of ordered pairs that can be substituted in to the system that will create true statements for BOTH equations
If you were to graph a system with one solution, the graphs of the two lines would intersect at a one single point. The graphs of the lines of system with no solution would never intersect with each other, meaning that the two lines would be parallel to each other. If the system has infinitely many solutions, then the graphs of the two lines would fall on top of one another, intersecting at all points along the two lines. Another way to think of this is by thinking about snails. Yep, snails. A snail went across the sidewalk in a straight line and made a trail. Another snail went across the same sidewalk in the exact same place as the first snail and also made a trail. At how many locations are the snails trails the same?
Let's take a look at what these special types of solutions would look like graphically.
What about when solving a system using substitution or linear combinations? What happens when you have a system that has no solution or infinitely many solutions? If a system of linear equations has no solution, all variable terms end up adding/subtracting out to zero, and you are left with a FALSE statement. If a system of linear equations has infinitely many solutions, all variable terms will add/subtract to zero, but you will be left with a TRUE statement. Look at the examples below.
Example 1: \(\begin{align*}&y=3x+5\\
&-6x+2y=10\end{align*}\) Substitution \(\begin{align}-6x+2(3x+5)&=10\\ -6x+6x+10&=10\\ 10&=10\end{align}\) True Statement |
Example 2: \(\begin{align*}2x+2y&=8\\x+y&=-4\end{align*}\)
Linear Combination \(\begin{align*}x(-2)+y(-2)&=-4(-2)\\ -2x-2y&=8\\ \underline{+2x+2y}&\underline{=8}\\ 0+0&=16\\ 0&{\neq}16\end{align*}\) False Statement |
In example 1, since the system ends with a true statement, the system should have infinitely many solutions. Since the system in example 2 ends with a false statement, this system would have no solution.
Quick Check
Solve the system to determine the number of solutions.
\( \begin{align*} & y= -4x-5 \\ & 8x+2y=-10 \\ \end{align*} \)
Solve the system to determine the number of solutions.
\( \begin{align*} & y= -4x-5 \\ & 8x+2y=-10 \\ \end{align*} \)