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# Graphs of Linear Systems

## Graph lines to identify intersection points

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Solving Linear Systems by Graphing
License: CC BY-NC 3.0

Two trains leave the station going in the same direction. One train leaves two hours before the other one. If the first train travels at an average speed of 65 mph and the second train travels at an average speed of 90 mph, then how long will it take the second train to catch up to the first train?

In this concept, you will learn to solve linear systems by graphing.

### Graphing Linear Systems

The solution for a system of two linear equations with two variables is an ordered pair that will make both equations true. If an ordered pair makes one of the linear equations true then the ordered pair represents a point on the line. If the same ordered pair makes the second linear equation true then the ordered pair represents a point on that line. If the same point is on the two lines, then the lines both pass through this same point. This means that the two lines will cross each other or intersect at this point.

A system of equations can have one solution, no solution or an infinite number of solutions.

#### Solve the following system of linear equations by graphing.

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

Notice that both equations are written in slope-intercept form \begin{align*}y=mx+b\end{align*}.

First, write down the information given in each equation.

\begin{align*}\begin{array}{rcl} y &=& mx+b \\ y &=& 2x-4 \\ m &=& 2; b = -4 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} y &=& mx+b \\ y &=& -2x+8 \\ m &=& -2; b=8 \end{array}\end{align*}

Next, graph each of the equations on the same Cartesian grid.

License: CC BY-NC 3.0

The two lines intersect at the point \begin{align*}(3,2)\end{align*}.

The solution is \begin{align*}\dbinom{x}{y} = \dbinom{3}{2}\end{align*}.

#### Solve the following system of linear equations by graphing.

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

Notice that both equations are written in slope-intercept form \begin{align*}y=mx+b\end{align*}.

First, write down the information given in each equation.

\begin{align*}\begin{array}{rcl} y &=& mx+b \\ y &=& 3x-1 \\ m &=& 3; b=-1 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} y &=& mx+b \\ y &=& 3x+2 \\ m &=& 3; b=2 \end{array}\end{align*}

The equations have the same slopes but different \begin{align*}y\end{align*}-intercepts.

Next, graph each of the equations on the same Cartesian grid.

License: CC BY-NC 3.0

The lines are parallel which means they will never intersect. There is no solution for this system of equations.

#### Solve the following system of linear equations by graphing.

\begin{align*}\begin{array}{rcl} y &=& - \frac{1}{4} x -1 \\ x+4y &=& -4 \end{array}\end{align*}

First, write down the information given in each equation.

\begin{align*}\begin{array}{rcl} y &=& mx+b \\ y &=& -\frac{1}{4}x-1 \\ m &=& -\frac{1}{4}; b=-1 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} x+4y &=& -4 \qquad \text{written in standard form} \ Ax+By=C \\ x- \text{intercept} &=& (-4,0) \\ y- \text{intercept} &=& (0,-1) \end{array}\end{align*}

The second equation is a multiple \begin{align*}(4)\end{align*} of the first equation.

Next, graph each of the equations on the same Cartesian grid.

License: CC BY-NC 3.0

The lines coincide which means they are actually the graph of the same line. There are an infinite number of solutions for this system of equations.

### Examples

#### Example 1

Earlier, you were given a problem about the two trains that left the station at different times. The second train left two hours after the first train but was travelling at a greater speed than the first train. You need to figure out when the second train will catch up to the first train.

First, write a system of linear equations to represent the distance travelled by each of the trains. Remember that distance \begin{align*}(d)\end{align*} is a function of the product of speed \begin{align*}(s)\end{align*} and time \begin{align*}(t)\end{align*}.

\begin{align*}\begin{array}{rcl} d &=& st\\ \text{Train }1: d &=& 65(t+2) \\ d &=& 65t+130 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} d &=& st \\ \text{Train } 2 : d &=& 90t \end{array}\end{align*}

Next, graph the two equations on the same Cartesian grid.

License: CC BY-NC 3.0

The lines display an intersection point just past five hours. This means the two trains will catch up with each other at this point.

Solve the following system of linear equations by graphing:

\begin{align*}\begin{array}{rcl} 2x+3y &=& 4 \\ x-5y &=& -11 \end{array}\end{align*}

Both equations are written in standard form \begin{align*}Ax+By=C\end{align*}.

First, write each equation in slope-intercept form \begin{align*}y=mx+b\end{align*} and write down the value of the slope and the \begin{align*}y\end{align*}-intercept.

\begin{align*}\begin{array}{rcl} y &=& mx+b \\ y &=& -\frac{2}{3}x+\frac{4}{3} \\ m &=& -\frac{2}{3}; b=\frac{4}{3} \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} y &=& mx+b \\ y &=& \frac{1}{5}x+\frac{11}{5} \\ m &=& \frac{1}{5}; b=\frac{11}{5} \end{array}\end{align*}

Then, graph both equations on the same Cartesian grid.

License: CC BY-NC 3.0

The two lines intersect at the point \begin{align*}(-1,2)\end{align*}. The solution for this system of linear equations is:

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

Answer the following examples using True or False. If the answer is false, explain why.

#### Example 2

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

The answer is true.

#### Example 3

If a system of linear equations is such that both equations have the same slope but different \begin{align*}y\end{align*}-intercepts then there are an infinite number of solutions for the linear system.

The answer is false.

If two equations have the same slope then the graph of the equations will display two parallel lines with different \begin{align*}y\end{align*}-intercepts. Parallel lines will never intersect. There is no solution for this linear system.

#### Example 4

If a system of linear equations is such that the two lines coincide then there is no solution for the linear system.

The answer is false.

If the graph of the linear system displays two lines that coincide then every ordered pair is on both lines and satisfy both equations. There is an infinite number of solutions for the linear system.

### Review

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

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

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

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

4. \begin{align*}\begin{array}{rcl} y &=& 2x \\ \frac{y}{2} &=& x - \frac{5}{2} \end{array}\end{align*}

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

6. \begin{align*}\begin{array}{rcl} y&=&2x+1 \\ y&=&3x+3 \end{array}\end{align*}

7. \begin{align*}\begin{array}{rcl} y&=&\frac{1}{2}x-2 \\ y&=&-2x \end{array}\end{align*}

8. \begin{align*}\begin{array}{rcl} y &=& -2x+1 \\ y&=&-2x-2 \end{array}\end{align*}

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

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

11. \begin{align*}\begin{array}{rcl} y&=&2x \\ y &=&3x-5 \end{array}\end{align*}

12. \begin{align*}\begin{array}{rcl} y&=&-x-1 \\ y&=&-x+6 \end{array}\end{align*}

Answer each question true or false.

13. Some linear systems do not have a solution.

14. Perpendicular lines have one solution.

15. Lines with the same slope and \begin{align*}y\end{align*}-intercept have infinite solutions.

To see the Review answers, open this PDF file and look for section 9.14.

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### Vocabulary Language: English

Consistent

A system of equations is consistent if it has at least one solution.

Dependent

A system of equations is dependent if every solution for one equation is a solution for the other(s).

Independent

A system of equations is independent if it has exactly one solution.

linear equation

A linear equation is an equation between two variables that produces a straight line when graphed.

Linear Function

A linear function is a relation between two variables that produces a straight line when graphed.

system of equations

A system of equations is a set of two or more equations.