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Graphs of Linear Equations

Graph lines presented in ax+by = c form

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Graph a Line in Standard Form

Scott and Brooke are organizing a fundraiser for their school. They are planning a pasta dinner, where adult tickets will cost $16 and kids' tickets will cost $8. Their goal is to make $2000. If they sell only adult tickets, how many must they sell to reach their goal? If they sell only kids' tickets, how many must they sell to reach their goal?

Graphing a Line in Standard Form

When a line is in standard form, there are two different ways to graph it. The first is to change the equation to slope-intercept form and then graph as shown in the previous concept. The second is to use standard form to find the \begin{align*}x\end{align*} and \begin{align*}y-\end{align*}intercepts of the line and connect the two. 

Graph the following equation

Graph \begin{align*}5x - 2y = -15\end{align*}.

Let’s use approach #1; change the equation to slope-intercept form.

\begin{align*}5x -2y &= -15\\ -2y &= -5x - 15\\ y &= \frac{5}{2}x + \frac{15}{2}\end{align*}

The \begin{align*}y-\end{align*}intercept is \begin{align*}\left( 0, \frac{15}{2} \right)\end{align*}. Change the improper fraction to a decimal and approximate it on the graph, (0, 7.5). Then use slope triangles. If you find yourself running out of room “rising 5” and “running 2,” you could also “fall 5” and “run backwards 2” to find a point on the other side of the \begin{align*}y-\end{align*}intercept.

Find the x and y intercepts of the equation \begin{align*}4x - 3x = 21\end{align*}.

Recall the Standard Form concept. The other coordinate will be zero at these points. Therefore, for the \begin{align*}x-\end{align*}intercept, plug in zero for \begin{align*}y\end{align*} and for the \begin{align*}y-\end{align*}intercept, plug in zero for \begin{align*}x\end{align*}.

\begin{align*}4x - 3(0) &= 21 && 4(0) -3y = 21\\ 4x &= 21 && \quad \ \ -3y = 21\\ x &= \frac{21}{4} \ or \ 5.25 && \qquad \quad \ y = -7\end{align*}

Graph the equation from previous problem

Use approach #2 from above. Plot each intercept from the previous problem on their respective axes and draw a line to connect them.

Examples

Example 1

Earlier, you were asked how many adult and kids' tickets must they sell to reach their goal.

The equation, in standard form, for the pasta dinner sales goal is \begin{align*}2000 = 16x + 8y\end{align*}. If they sell only adult tickets, we are looking for the x-intercept, so set y equal to zero.

\begin{align*}2000 = 16x + 8(0)\\ 2000 = 16x\\ x = 125\end{align*}

Therefore, they must sell 125 adult tickets to reach their goal.

If they sell only kids' tickets, we are looking for the y-intercept, so set x equal to zero.

\begin{align*}2000 = 16(0) + 8y\\ 2000 = 8y\\ y = 250\end{align*}

Therefore, they must sell 250 kids' tickets to reach their goal.

Example 2

Graph \begin{align*}4x + 6y = 18\end{align*} by changing it into slope-intercept form.

Change \begin{align*}4x + 6y = 18\end{align*} into slope-intercept form by solving for \begin{align*}y\end{align*}, then graph.

\begin{align*}4x +6y &= 18\\ 6y &= -4x + 18\\ y &= - \frac{2}{3}x + 3\end{align*}

Example 3

Graph \begin{align*}5x - 3y = 30\end{align*} by plotting the intercepts.

Substitute in zero for \begin{align*}x\end{align*}, followed by \begin{align*}y\end{align*} and solve each equation.

\begin{align*}5(0) - 3y &= 30 && 5x -3(0) = 30\\ -3y &= 30 && \qquad \quad 5x = 30\\ y &= -10 && \qquad \quad \ x = 6\end{align*}

Now, plot each on their respective axes and draw a line.

Review

Graph the following lines by changing the equation to slope-intercept form.

  1. \begin{align*}-2x + y = 5\end{align*}
  2. \begin{align*}3x + 8y = 16\end{align*}
  3. \begin{align*}4x -2y = 10\end{align*}
  4. \begin{align*}6x + 5y = -20\end{align*}
  5. \begin{align*}9x - 6y = 24\end{align*}
  6. \begin{align*}x + 4y = -12\end{align*}

Graph the following lines by finding the intercepts.

  1. \begin{align*}2x + 3y = 12\end{align*}
  2. \begin{align*}-4x + 5y = 30\end{align*}
  3. \begin{align*}x - 2y = 8\end{align*}
  4. \begin{align*}7x + y = -7\end{align*}
  5. \begin{align*}6x + 10y = 15\end{align*}
  6. \begin{align*}4x -8y = -28\end{align*}
  7. \begin{align*}y=3\end{align*}
  8. Writing Which method do you think is easier? Why?
  9. Writing Which method would you use to graph \begin{align*}x = -5\end{align*}? Why?

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.7. 

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Vocabulary

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.

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

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