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## Algebra basics (ASL)

### Course: Algebra basics (ASL) > Unit 5

Lesson 4: Number of solutions to systems of equations (ASL)- Systems of equations number of solutions: fruit prices (1 of 2)
- Systems of equations number of solutions: fruit prices (2 of 2)
- Solutions to systems of equations: consistent vs. inconsistent
- Solutions to systems of equations: dependent vs. independent
- Number of solutions to a system of equations
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations algebraically
- Number of solutions to a system of equations algebraically
- How many solutions does a system of linear equations have if there are at least two?
- Number of solutions to system of equations review

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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.

### Example system with one solution

We're asked to find the number of solutions to this system of equations:

Let's put them in slope-intercept form:

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:

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:

Interestingly, if we multiply the second equation by minus, 2, we get the first equation:

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

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