Do you know how to identify a linear system of equations?

If you think about it, you are already familiar with a linear equation.

\begin{align*}2x+3=11\end{align*}

Here is a linear equation. When you have a linear equation, you can simply solve it for \begin{align*}x\end{align*}.

You can also have an equation where two variables are present.

\begin{align*}2x+y=10\end{align*}

This is a linear equation in standard form.

What about a system of linear equations? Do you know how to identify one? Do you know how to solve one?

**This Concept will show you how to work with linear systems. You will be able to answer these questions by the end of the Concept.**

### Guidance

Linear functions are useful all by themselves, and yet there are other applications of the idea. In a system of linear equations, you will see how linear equations can work together in a system to solve even more complex problems. Indeed, there are various ways that we can find solutions to these problems or find that there may be no solution at all.

If you add two numbers together, you get 13. Can you think of any ordered pairs that would work for this?

(1, 12), (3, 10), (-4, 17), (4.5, 8.5)

Hopefully you’ll agree that there are infinite pairs of numbers whose sum is 13. You might also agree that there are infinite pairs of numbers whose difference is 7.

(9, 2), (11, 4), (37, 30), (95.8, 88.8), (-3, -10)

However, which ordered pair is true for both conditions at once? Which pair has a sum of 13 and a difference of 7?

If you make a list of ordered pairs, you can check them to see which makes both equations true.

This is a *system of equations***—two or more equations at the same time.**

In the situation above, the solution is (10, 3) because the sum of the two numbers is 13 and their difference is 7.

**The pair (10, 3) makes both equations true.**

A solution to a system of equations is an ordered pair that makes both equations true. Is there always a solution? Can there be more than one solution? Let’s investigate this.

Two numbers have a sum of 17. If you add two numbers together, their sum is 15. As you know, there are infinite ordered pairs whose sum is 17. There are also infinite ordered pairs whose sum is 15. But can a single ordered pair have a sum of both 17 and 15 at the same time?

**First, let’s write two equations to help us to sort out the information in this system of equations. There are two equations and both have a different sum.**

\begin{align*} x+y &=17 \\ x+y &=15\end{align*}

**If we think about these two equations, you will see that there aren’t any values that will work for both of these equations.**

**Therefore, this system has no solutions.**

Here is another one.

Two numbers have a sum of -8. Twice the first number plus twice the second number is -16.

**First, let’s write the two equations described above. Then we can investigate possible solutions.**

\begin{align*} x+y &= -8 \\ 2x+2y &=-16 \end{align*}

**Does this system have a solution? Think of a solution for the first equation. How about (-3, -5)? Does it work for the second? Yes. Think of another solution like (9, -1). This one is also true in both equations.**

**This equation has an infinite number of solutions.**

**Some systems of equations have infinite solutions because all ordered pairs that make one equation true also make the other true.**

Answer each question true or false.

#### Example A

A linear system is two equations where the value of \begin{align*}x\end{align*} is the solution for the system.

**Solution: False**

#### Example B

The solution for a linear system is written as an ordered pair.

**Solution: True**

#### Example C

Some linear systems do not have a solution.

**Solution: True**

Now let's go back to the dilemma from the beginning of the Concept.

Identifying a linear system means that you will see two equations where there are unknown values for both \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

Solving a linear system requires you to find two values that will work as the values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in both equations.

This solution is then written as an ordered pair.

If you can't find a solution, then the system does not have a solution.

### Vocabulary

- System of Equations
- two or more equations at the same time. The solution will be the ordered pair that works for both equations.

### Guided Practice

Here is one for you to try on your own.

Which ordered pair makes both equations true?

1. \begin{align*}x+y &=8 \\ 4x-y &=-3\end{align*}

a. (2, 6)

b. (3, 15)

c. (4, 4)

d. (1, 7)

Let’s test each pair and see which pair, if any, works:

a. \begin{align*}x+y &=8 \\ 2+6 &=8? \\ 8 &=8 \\ 4x-y &=-3 \\ 4 \cdot 2-6 &=-3? \\ 8-6 &=-3? \\ 2 &\ne -3 \end{align*}

**Solution**

The ordered pair (2, 6) makes the first equation true, but not the second. Because it is not true for both equations, it is not a solution to the system.

b. \begin{align*}x+y &=8 \\ 3+15 &=8? \\ 18 &\ne 8 \end{align*}

The ordered pair (3, 15) does not even make the first equation true. It cannot be a solution to the system.

c. \begin{align*}x+y &=8 \\ 4+4 &=8? \\ 8 &=8 \\ 4x-y &=-3 \\ 4 \cdot 4-4 &=-3? \\ 16-4 &=-3 \\ 12 &\ne -3\end{align*}

The ordered pair (4, 4) makes the first equation true, but not the second. Because it is not true for both equations, it is not a solution to the system.

d. \begin{align*}x+y &=8 \\ 1+7 &=8? \\ 8 &=8 \\ 4x-y &=-3 \\ 4 \cdot 1-7 &=-3? \\ 4-7 &=-3?\\ -3 & = -3\end{align*}

**The ordered pair (1, 7) makes both equations true. This is a solution to the system.**

### Video Review

### Practice

Directions: Figure out which pair is a solution for each given system.

- Which ordered pair is a solution of the following system?

- \begin{align*} x-3y &=9 \\ 3x+y &=7\end{align*}
- (a) \begin{align*}(6, -1)\end{align*}
- (b) \begin{align*}(-1, -4)\end{align*}
- (c) \begin{align*}(0, 7)\end{align*}
- (d) \begin{align*}(3, -2)\end{align*}

- Which ordered pair is a solution of the following system?

- \begin{align*} y &=3x-7 \\ 5x-3y &=13\end{align*}
- (a) \begin{align*} \left (3, \frac{2}{3} \right) \end{align*}
- (b) \begin{align*}(2, -1)\end{align*}
- (c) \begin{align*}(4, 7)\end{align*}
- (d) \begin{align*}(5, 8)\end{align*}

Directions: Determine whether each system has infinite solutions or no solutions.

- .

- \begin{align*} x +y &= 10 \\ y &= -x +10 \end{align*}

- .

- \begin{align*} 3x -6y &= -24 \\ x -2y &= -8 \end{align*}

- .

- \begin{align*} \frac{3}{4}x &= \frac{2}{3}y-1 \\ 9x &= 8y-12 \end{align*}

- .

- \begin{align*} y &= \frac{1}{2}x + 3 \\ y &= \frac{1}{2}x - 2 \\ \\ y &= 3x-5 \\ y &= 3x-2 \end{align*}

- .

- \begin{align*} y &= \frac{1}{2}x + 3 \\ y &= \frac{1}{2}x - 2 \end{align*}

Directions: Answer each question true or false.

- Parallel lines have the same slope.
- A linear system of equations cannot be graphed on the coordinate plane.
- Parallel lines have infinite solutions.
- Perpendicular lines have one solution.
- Lines with an infinite number of solutions are not parallel.
- Some linear systems do not have a solution.
- To solve a linear system, you must have a value for x and y.
- An ordered pair is never a solution for a linear system.