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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
• Current
How many solutions does the system of linear equations have?
\begin{aligned} y &= -2x+4\\\\ 7y &= -14x+28 \end{aligned}