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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.
A coordinate plane. The x- and y-axes both scale by one-half. A graph of a line goes through the points negative one-half, three and three, two. A graph of another line goes through the points zero, zero and one, one. These two lines intersect at an x-value between two and three and a y-value between two and three.
One solution. A system of linear equations has one solution when the graphs intersect at a point.
A coordinate plane. The x- and y-axes both scale by one-half. A graph of a line goes through the points one, one and a half and three, one. A graph of another line goes through the points one, two and a half and three, two. These two lines never intersect.
No solution. A system of linear equations has no solution when the graphs are parallel.
A coordinate plane. The x- and y-axes both scale by one-half. A graph of a line goes through the points zero, one and a half and three, two. A graph of another line goes through the points zero, one and a half and three, two. These lines overlap entirely.
Infinite solutions. A system of linear equations has infinite solutions when the graphs are the exact same line.
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{aligned} y&=-6x+8\\\\ 3x+y&=-4 \end{aligned}
Let's put them in slope-intercept form:
\begin{aligned} y&=-6x+8\\\\ y&=-3x-4 \end{aligned}
Since the slopes are different, the lines must intersect. Here are the graphs:
A coordinate plane. The x- and y-axes both scale by one-half. The equation y equals negative six x plus eight is graphed going through the points zero, eight and one, two. The equation three x plus y equals negative four is graphed going through the points zero, negative four and one, negative seven. These lines intersect at a value that is below the graph.
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{aligned} y &= -3x+9\\\\ y &= -3x-7 \end{aligned}
Without graphing these equations, we can observe that they both have a slope of minus, 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{aligned} -6x+4y &= 2\\\\ 3x-2y &= -1 \end{aligned}
Interestingly, if we multiply the second equation by minus, 2, we get the first equation:
\begin{aligned} 3x-2y &= -1\\\\ \blueD{-2}(3x-2y)&=\blueD{-2}(-1)\\\\ -6x+4y &= 2 \end{aligned}
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{aligned} y &= -2x+4\\\\ 7y &= -14x+28 \end{aligned}

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!
• 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.
(1 vote)
• How does the quadratic formula come from? How to prove this formula ?
• Can anybody explain me ...........in the 1st equation which is Y=-6x+8 ............i know it`s from y intercept chapter.....How do you draw on graph ?on this equation 6 is slope and 8 is the value of y......so 8+ goes to y+ graph but from where did you get other point ........it looks like more than 1.....How do you figure it out ?
• ok, so first off, the slope is -6 not 6 because of the negative sign in front of the -6x. And then you use the 8 as the intercept. as a result, one of the points become (0,8). Then, you go six down and one to the right. That's another point and you can keep doing that.
(1 vote)
• What is algebra 1 question
• If you're relating to this question;
y = −6x+8
y = −3x−4
First step: Either use substitution or elimination to find where they intersect. For this, I will use substitution.
First, rearrange an equation so that you can substitute a value in; y = -6x + 8 -> x = -y/6 + 4/3. From there you substitute into any of your equations and you find y afterwards. I hope this helped!

• I also want to learn how to make some extremely hard hitting and brutal Deathstep basses in a program I bought called Serum by XFer records.
• How should write a system of equations that has the solution of (2,-3)?