Suppose that there is a linear relationship between your annual income and the amount you must pay in state income tax. What if you wanted to make a graph that showed how much tax you must pay based on your income? Could you do it?

### Graphing Linear Equations

Previously, you learned how to solve equations in one variable. The answer was of the form "variable = some number". Now, we are going to talk about two-variable equations. Below are several examples of two-variable equations:

\begin{align*}p& =20(h)\\ m& =8.15(n)\\ y& =4x+7\end{align*}

Their solutions are not one value because there are two variables. The solutions to an equation with two variables are sets of ordered pairs and these pairs of numbers can be graphed in a Cartesian plane.

A **linear equation** is a n equation whose graph is a straight line. The solutions to such an equation are all the coordinates on the graphed line.

By making a table, you are finding the solutions to the equation with two variables.

#### Let's graph the following linear equations:

- Graph \begin{align*}p=2(h)\end{align*}

Make a table and then graph the points:

\begin{align*}h\end{align*} |
\begin{align*}p\end{align*} |
---|---|

0 | 0 |

1 | 2 |

2 | 4 |

3 | 6 |

- \begin{align*} y=4x+7\end{align*}

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

-1 | 3 |

0 | 7 |

1 | 13 |

- A taxi fare costs more the further you travel. Taxis usually charge a fee on top of the per-mile charge. In this case, the taxi charges $3 as a set fee and $0.80 per mile traveled. Find all the possible solutions to this equation.

The equation linking the cost in dollars \begin{align*}(y)\end{align*} to hire a taxi and the distance traveled in miles \begin{align*}(x)\end{align*} is: \begin{align*}y=0.8x+3\end{align*}.

This is an equation in two variables. By creating a table, we can graph these ordered pairs to find the solutions.

\begin{align*}x\end{align*} (miles) |
\begin{align*}y\end{align*} (cost $) |
---|---|

0 | 3 |

10 | 11 |

20 | 19 |

30 | 27 |

40 | 35 |

The solutions to the taxi problem are located on the green line graphed above. To find any cab ride cost, you just need to find the \begin{align*}y\end{align*} of the desired \begin{align*}x\end{align*}.

### Examples

#### Example 1

Earlier, you were told to suppose that there is a linear relationship between your annual income and the amount you must pay in state income tax. How would you make a graph to show how much tax you must pay based on your income?

As shown in this Concept, to graph this relationship, first you would need a linear equation that represents the relationship. After that, use a table to find points to plot and then graph the points. Finally, connect the points with a line. The line demonstrates how much tax you must pay with the \begin{align*}x\end{align*}-value representing your income and the \begin{align*}y\end{align*}-value representing the tax you must pay.

#### Example 2

Graph \begin{align*}m=8.15(n)\end{align*}.

To graph this linear equation, we will make a table of some points. We can plug in values for \begin{align*}h\end{align*} to get values for \begin{align*}m\end{align*}. For example:

\begin{align*}m=8.15(-1)=-8.15\end{align*}

\begin{align*}n\end{align*} |
\begin{align*}m\end{align*} |
---|---|

\begin{align*}-1\end{align*} | \begin{align*}-8.15\end{align*} |

\begin{align*}1\end{align*} | \begin{align*}8.15\end{align*} |

\begin{align*}2\end{align*} | \begin{align*}16.3\end{align*} |

Next, we will graph each point, and then connect the points with a line.

### Review

- What are the solutions to an equation in two variables? How is this different from an equation in one variable?
- Think of a number. Triple it, and then subtract seven from your answer. Make a table of values and plot the equation that this sentence represents.

Graph the solutions to each linear equation by making a table and graphing the coordinates.

- \begin{align*}y=2x+7\end{align*}
- \begin{align*}y=0.7x-4\end{align*}
- \begin{align*}y=6-1.25x\end{align*}

**Mixed Review**

- Find the sum: \begin{align*}\frac{3}{8}+\frac{1}{5}-\frac{5}{9}\end{align*}.
- Solve for \begin{align*}m: 0.05m+0.025(6000-m)=512\end{align*}.
- Solve the proportion for \begin{align*}u: \frac{16}{u-8}=\frac{36}{u}\end{align*}.
- What does the Additive Identity Property allow you to do when solving an equation?
- Shari has 28 apples. Jordan takes \begin{align*}\frac{1}{4}\end{align*} of the apples. Shari then gives away 3 apples. How many apples does Shari have?
- The perimeter of a triangle is given by the formula \begin{align*}Perimeter=a+b+c\end{align*}, where \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*} are the lengths of the sides of a triangle. The perimeter of \begin{align*}\triangle ABC\end{align*} is 34 inches. One side of the triangle is 12 inches. A second side is 7 inches. How long is the remaining side of the triangle?
- Evaluate \begin{align*}\frac{y^2-16+10y+2x}{2}\end{align*}, for\begin{align*}x=2\end{align*}and\begin{align*}y=-2.\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.2.