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# Number of solutions to system of equations review

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line). This article reviews all three cases.
Want to learn more about the number of solutions to systems of equations? Check out this video.

### Example system with one solution

We're asked to find the number of solutions to this system of equations:
$\begin{array}{rl}y& =-6x+8\\ \\ 3x+y& =-4\end{array}$
Let's put them in slope-intercept form:
$\begin{array}{rl}y& =-6x+8\\ \\ y& =-3x-4\end{array}$
Since the slopes are different, the lines must intersect. Here are the graphs:
Because the lines intersect at a point, there is one solution to the system of equations the lines represent.

### Example system with no solution

We're asked to find the number of solutions to this system of equations:
$\begin{array}{rl}y& =-3x+9\\ \\ y& =-3x-7\end{array}$
Without graphing these equations, we can observe that they both have a slope of $-3$. This means that the lines must be parallel. And since the $y$-intercepts are different, we know the lines are not on top of each other.
There is no solution to this system of equations.

### Example system with infinite solutions

We're asked to find the number of solutions to this system of equations:
$\begin{array}{rl}-6x+4y& =2\\ \\ 3x-2y& =-1\end{array}$
Interestingly, if we multiply the second equation by $-2$, we get the first equation:
$\begin{array}{rl}3x-2y& =-1\\ \\ -2\left(3x-2y\right)& =-2\left(-1\right)\\ \\ -6x+4y& =2\end{array}$
In other words, the equations are equivalent and share the same graph. Any solution that works for one equation will also work for the other equation, so there are infinite solutions to the system.

## Practice

Problem 1
How many solutions does the system of linear equations have?
$\begin{array}{rl}y& =-2x+4\\ \\ 7y& =-14x+28\end{array}$

Want more practice? Check out these exercises:

## Want to join the conversation?

• Greetings, may we use systems of equations to solve real world problems?
• Yes - the one that used to be used a lot was to compare cell phone plans, but now that many are unlimited data, it is not the same. Economics uses it for example in profit loss graphs. Engineers still use systems. It is used for comparing mortgages and other comparisons.
• can we find a systeme of equation that has infinite solution apply in the real life ??
• Gwyndolin wants to buy some tomatoes and two vegetation vendors happen to be selling tomatoes. Stall A sells them for $0.50 each and stall B sells them for 2 for a$1.00. both stalls charge a \$0.10 visit fee.
stall A) y = 0.50x + 0.10
stall B) y = 1/2x + 0.10
• How do u know where to put the lines?
• For each equation, you need to find at least two points. Then you draw a line that goes through the points.

The easiest way to find the points is to just pick a value for one of the variables (x or y) and then solve for the other variable.

For instance, take the equation y = -2x + 4

Let's say x = 0
y = -2x + 4
y = -2(0) + 4
y = 0 + 4
y = 4
So when x = 0, y = 4
Our first point is (0, 4)

Now, let's say x = 2
y = -2x + 4
y = -2(2) + 4
y = -4 + 4
y = 0
So when x = 2, y = 0
Our second point is (2, 0)

Finally, let's say x = 1
y = -2x + 4
y = -2(1) + 4
y = -2 + 4
y = 2
So when x = 1, y = 2
Our third point is (1, 2)

Now, just graph these three points on paper
(0, 4)
(2, 0)
(1, 2)
and then draw a line that goes through all of the points.

For the second equation, repeat the process to find the points and then graph the points / line on the same xy graph as the first equation.

You will know where to put the lines by finding the points for each equation.

Hope this helps!
• what is (y1-y)=m(x1-x)
• That is the point slope form of a linear equation multiplied by -1. Point slope form is y-y1=m(x-x1)
• How do you find the solution to a system without using a graph? Is it even possible to do that?
• How would you solve:
y = -2x + 1
y = 4x - 3
It doesn't make sense. Thanks!
• If y = -2x + 1
and y is also 4x - 3
then -2x + 1 = 4x - 3
so -6x = -4
so x = 4/6 = 2/3
Now that you know the value of x, you can plug it back into the equations to solve for y.
• it the year 2023
• What about non-linear systems of equations like
2x^2+3x-4
-0.5x^2-x+1
Could they have exactly 2 solutions?
• Yes! If you equate them to each other, you'll get a third quadratic in x. Then, solving that will give you the two x coordinates of their intersection, thus proving that they do indeed have two solutions.

Do note that a system of two quadratics can have a single solution too. But at max, they'll have two (unique) ones.