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# Graphs of Linear Equations

## Graph lines presented in ax+by = c form

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Practice Graphs of Linear Equations
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Using Tables to Graph Functions

Do you know how to graph an equation? Take a look at this dilemma.

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

Here is an equation. This equation is in function form and can be graphed. Do you know how to do this? Is this a linear equation?

This Concept will show you how to graph and identify different equation. You will see this one again at the end of the Concept.

### Guidance

Do you know how to graph an equation from a table of values? Well, once you have a table of values that has been created from a function or an equation, you can take the values in the table and graph them.

Take a look.

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

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 1
1 5
2 9
3 13

Now we have an equation in function form. We have a table of values that makes this equation true, so we can look at graphing this function.

First, let’s write out the ordered pairs from the table. Notice that these values are all positive, however, you can have positive and negative values that make an equation true.

(0, 1)

(1, 5)

(2, 9)

(3, 13)

Notice that the graph of this equation forms a straight line. When a set of values are graphed to represent an equation, if a straight line is created, we call this a linear equation. Linear means line.

Now you know how to graph a linear equation and how to identify this equation based on the graph. What about an equation like \begin{align*}y=5\end{align*} or \begin{align*}x=-3?\end{align*} These are equations we don’t need to make tables. If \begin{align*}y\end{align*} is equal to five, then we could say that \begin{align*}x\end{align*} is equal to all of the other values.

Now let’s look at the graph of this equation.

You can see that the graph of this line is a horizontal line.

Next, we can look at the equation \begin{align*}x= -3\end{align*}. Now we can graph this line. If we know that the \begin{align*}x\end{align*} value is -3, then we can say that all of the other \begin{align*}y\end{align*} values are all of the other numbers. Let’s take a look at this graph.

The graph of the equation \begin{align*}x = -3\end{align*} is a vertical line.

We can say that when \begin{align*}x\end{align*} is equal to one value, then we have a vertical line and when \begin{align*}y\end{align*} is equal to one value that we have a horizontal line.

Write this information down in your notebook. Be sure that you understand how to identify the graph of a horizontal or a vertical line.

Describe each line.

#### Example A

\begin{align*}x = 5\end{align*}

Solution: Vertical line at positive 5

#### Example B

\begin{align*}y = -2\end{align*}

Solution: Horizontal line at negative 2

#### Example C

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

Solution: A straight line that is slanted going through point (0,1) on the y-axis.

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

First, notice that this equation is already in function form, so we can create a table of values. This table will give us our ordered pairs for graphing.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 -1
1 1
2 3
3 5

Here are the ordered pairs.

(0, -1)

(1, 1)

(2, 3)

(3, 5)

Now let’s graph the equation.

You can see from the graph that this is a linear equation. The graph of the line is a straight line.

### Vocabulary

Ordered pair
the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values that can be found in a table or used to graph points or a line on the coordinate plane.
Standard form
the form of an equation \begin{align*}Ax+By=C\end{align*}
Function form
the form of an equation \begin{align*}y=mx+b\end{align*}
Linear Equation
the graph of a linear equation forms a straight line.

### Guided Practice

Here is one for you to try on your own.

Use this graph to create a table of values.

Solution

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
1 11
0 7
-1 3

### Practice

Directions: Create a table of values for each equation and then graph it on the coordinate plane.

1. \begin{align*}y=2x+1\end{align*}
2. \begin{align*}y=3x+2\end{align*}
3. \begin{align*}y=-4x\end{align*}
4. \begin{align*}y=-2x\end{align*}
5. \begin{align*}y=-3x+3\end{align*}
6. \begin{align*}y=2x+3\end{align*}
7. \begin{align*}y=3x-2\end{align*}
8. \begin{align*}y=-8x\end{align*}
9. \begin{align*}y=3x+1\end{align*}
10. \begin{align*}y=4x\end{align*}
11. \begin{align*}y=-2x+2\end{align*}
12. \begin{align*}y=2x-2\end{align*}
13. \begin{align*}y=x-1\end{align*}
14. \begin{align*}x=4\end{align*}
15. \begin{align*}y=-2\end{align*}

### Vocabulary Language: English

Cartesian Plane

Cartesian Plane

The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.
Function

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
linear equation

linear equation

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

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$
Standard Form

Standard Form

The standard form of a line is $Ax + By = C$, where $A, B,$ and $C$ are real numbers.