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Intercepts by Substitution

Graphing equations that form lines by finding x and y intercepts

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Intercepts by Substitution

Suppose the number of gallons of gas left in the tank of a car depends on the number of miles driven and can be represented by a linear equation. You know how much gas is in the tank when 0 miles have been driven, and you also know after how many miles you will have 0 gallons left in the tank. Could you graph the linear equation? 

Finding Intercepts by Using Substitution

Graphing solutions to an equation of two variables using a table can be time-consuming. Fortunately, there are several ways to make graphing solutions easier. This Concept will focus on graphing a line by finding its intercepts

In geometry, there is a postulate that states, “Two points determine a line.” Therefore, to draw any line, all you need is two points. One way is to find its intercepts.

An intercept is the point at which a graphed equation crosses an axis.

The \begin{align*}x-\end{align*}intercept is an ordered pair at which the line crosses the \begin{align*}x-\end{align*}axis (the horizontal axis). Its ordered pair has the form \begin{align*}(x,0)\end{align*}.

The \begin{align*}y-\end{align*}intercept is an ordered pair at which the line crosses the \begin{align*}y-\end{align*}axis (the vertical axis). Its ordered pair has the form \begin{align*}(0,y)\end{align*}.

By finding the intercepts of an equation, you can quickly graph all the possible solutions to the equation.

Remember that the Substitution Property allows the replacement of a variable with a numerical value or another expression. You can use this property to help find the intercepts of an equation.

Let's graph the following linear equations using its intercepts:

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

The \begin{align*}x-\end{align*}intercept has the ordered pair \begin{align*}(x,0)\end{align*}. Therefore, the \begin{align*}y-\end{align*}coordinate has a value of zero. By substituting zero for the variable of \begin{align*}y\end{align*}, the equation becomes:

\begin{align*}2x+3(0)=-6\end{align*}

Continue solving for \begin{align*}x\end{align*}:

\begin{align*}2x+0& =-6\\ 2x& =-6\\ x& =-3\end{align*}

The \begin{align*}x-\end{align*}intercept has an ordered pair of (–3, 0).

Repeat the process to find the \begin{align*}y-\end{align*}intercept. The ordered pair of the \begin{align*}y-\end{align*}intercept is \begin{align*}(0,y)\end{align*}. Using substitution:

\begin{align*}2(0)+3y& =-6\\ 3y& =-6\\ y& =-2\end{align*}

The \begin{align*}y-\end{align*}intercept has the ordered pair (0, –2).

To graph the line formed by the solutions of the equation \begin{align*}2x+3y=-6\end{align*}, graph the two intercepts and connect them with a straight line.

  1. \begin{align*}4x-2y=8\end{align*}

Determine the \begin{align*}x-\end{align*}intercept by substituting zero for the variable \begin{align*}y\end{align*}.

\begin{align*}4x-2(0)& =8\\ 4x& =8\\ x& =2\end{align*}

The ordered pair of the \begin{align*}x-\end{align*}intercept is (2, 0). By repeating this process, you'll find the \begin{align*}y-\end{align*}intercept has the ordered pair (0, –4). Graph these two ordered pairs and connect them with a line.

  1. \begin{align*}x-2y=10\end{align*}

First, find the intercepts by substituting in zero for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}:

\begin{align*} \text{Substitute in zero for x.} && (0)-2y &=10 \\ \text{Simplify.} && -2y&=10 \\ \text{Solve for y.}&& y&=-5 \end{align*}

This means that the \begin{align*}y\end{align*}-intercept is at the point (0,-5).

\begin{align*} \text{Substitute in zero for y.} && x-6(0) &=10 \\ \text{Solve for x.}&& x&=10 \end{align*}

This means that the \begin{align*}x\end{align*}-intercept is at the point (10,0).

   

 

 

Examples

Example 1

Earlier, you were asked to suppose that the number of gallons left in the tank of a car depends on the number of miles driven and can be represented by a linear equation. Could you graph the linear equation if you know how much gas is in the tank when 0 miles have been driven and how many miles it will take to have 0 gallons left in the tank?

Yes, you can draw a line with any two points. In this case, the amount of gas in the tank when 0 miles have been driven will be the \begin{align*}y-\end{align*}intercept while the number of miles it will take to have 0 gallons left in the tank will be the \begin{align*}x-\end{align*}intercept. With these two points, you can draw the line that represents the linear equation.

Example 2

Find the intercepts and use them to graph \begin{align*}y=-2x+8\end{align*}.

Substitute in zero for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, in order to find the intercepts.

\begin{align*} \text{Substitute in zero for x.} && y&=-2(0)+8 \\ \text{Solve for y.}&& y&=8 \end{align*}

This means that the \begin{align*}y\end{align*}-intercept is at the point (0,8).

\begin{align*} \text{Substitute in zero for y.} && 0&=-2x+8 \\ \text{Solve for x.}&& x&=4 \end{align*}

This means that the \begin{align*}x\end{align*}-intercept is at the point (4,0).

Review

  1. Define intercept.
  2. What is the ordered pair for an \begin{align*}x-\end{align*}intercept?

Find the intercepts for the following equations using substitution.

  1. \begin{align*}y=3x-6\end{align*}
  2. \begin{align*}y=-2x+4\end{align*}
  3. \begin{align*}y=14x-21\end{align*}
  4. \begin{align*}y=7-3x\end{align*}
  1. Which intercept does a vertical line have?
  2. Does the equation \begin{align*}y=5\end{align*} have both an \begin{align*}x-\end{align*}intercept and a \begin{align*}y-\end{align*}intercept? Explain your answer.
  3. Write an equation having only an \begin{align*}x-\end{align*}intercept at (–4, 0).
  4. How many equations can be made with only one intercept at (0, 0)? Hint: Draw a picture to help you.

Mixed Review

For 11-12, determine whether each ordered pair is a solution to the equation.

  1. \begin{align*}5x+2y=23;(7,-6)\end{align*}and (3, 4)
  2. \begin{align*}3a-2b=6;(0,3)\end{align*}and\begin{align*}\left (\frac{5}{3},\frac{-1}{2}\right )\end{align*}.
  3. Graph the solutions to the equation \begin{align*}x=-5\end{align*}.
  4. Solve: \begin{align*}\frac{4}{5} k-16=-\frac{1}{4}\end{align*}.
  5. Is the following relation a function? \begin{align*}\left \{(-1,1),(0,0),(1,1),(2,3),(0,6)\right \}\end{align*}
  6. Using either a finite set of counting numbers, whole numbers, or integers, how would you describe the domain of the following situation: The number of donuts purchased at a coffee shop on a particular day?
  7. Find the percent of change: Old price = $1,299; new price = $1,145.

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.5. 

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Vocabulary

X-intercept

An ordered pair at which the line crosses the x-axis (the horizontal axis). Its ordered pair has the form (x,0).

Y-intercept

The y-intercept is an ordered pair at which the line crosses the y-axis (the vertical axis). Its ordered pair has the form (0,y).

Intercept

The intercepts of a curve are the locations where the curve intersects the x and y axes. An x intercept is a point at which the curve intersects the x-axis. A y intercept is a point at which the curve intersects the y-axis.

Intercept Method

The intercept method is a way of graphing a linear function by using the coordinates of the x and y-intercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and then joining them with a straight line.

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