Have you ever seen a dilemma like this one? Take a look.

Two trains leave the station going in the same direction. One train leaves two hour before the other. The first train’s average speed is 65mph while the second train’s average speed is 90mph. How long will it take for the second train to catch up to the first?

**The two trains can represent a linear system. You can solve this linear system by graphing. You will learn how to solve it in this Concept.**

### Guidance

When we have a system of equations, we can graph the equations and solve the system by graphing. When you graph two linear equations, two lines are formed. These two lines will intersect. The point where the lines intersect is the solution for the system.

Take a look.

Solve the following system by graphing.

\begin{align*} y &= 2x-4 \\ y &= -2x+8\end{align*}

**To work on this system, we will graph both of these lines and look for the point of intersection. That point of intersection will be the solution to the system.**

**The point of intersection is (3, 2).**

**This is the solution to the system of equations.**

**Sometimes you might have two equations that do not intersect. If this happens, then you know that this is a system with no solutions. Parallel lines are one possible system that would not have a solution. Parallel lines have the same slope, so you can recognize a system with no solutions.**

Answer each question true or false.

#### Example A

If two lines intersect at a point, then the coordinates of that point are a solution to the linear system.

**Solution: True**

#### Example B

Parallel lines have the same slope.

**Solution: True.**

#### Example C

Perpendicular lines also have the same slope.

**Solution: False.**

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

**First, let’s think about functions with regard to this problem. The second train’s distance \begin{align*}d\end{align*} is a function of time \begin{align*}t\end{align*}. It can be found with the equation \begin{align*} d=90t \end{align*}.**

**Since the first train left two hours before the first train, it had two hours more than the second train to travel. Its speed was 65mph, though. Its distance can be found with the equation \begin{align*}d=65(t+2)\end{align*}.**

Solve the system of equations using a graph.

**From the graph you can see that the trains will meet shortly after 5 hours.**

### 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.

Does this graph show a solution for the following system? Why or why not?

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

**Solution**

This graph does not show a solution for the following system. The solution (2,1) works for the second equation, but not for the first equation.

**An ordered pair must be a solution for both equations in a system.**

### Video Review

Solving Linear Systems by Graphing

### Practice

Directions: Graph the following systems of equations. Identify the solution or write no solution or infinitely many solutions, if appropriate.

- .

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

- .

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

- .

- \begin{align*} y &= -3x + 1 \\ 3x - y &= -7 \end{align*}

- .

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

- .

- \begin{align*} y &= 3x+5 \\ y &= - 3x+8 \end{align*}

- .

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

- .

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

- .

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

- .

- \begin{align*} y &= 4x-1 \\ y &= 2x+2\end{align*}

- .

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

- .

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

- .

- \begin{align*} y &= -x-1 \\ y &= -x+6\end{align*}

Directions: Answer each question true or false.

- Some linear systems do not have a solution.
- Perpendicular lines have one solution.
- Lines with the same slope and y-intercept have infinite solutions.