2.2: Finding the Equation of a Line in Slope-Intercept Form
You decide to buy a laptop for $800. In 3 years, the laptop will be worth $450. How much will the computer be worth after 6 years?
Writing a linear equation that relates the two prices will help you determine how much the computer will be worth after 6 years.
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James Sousa: Slope Intercept Form of a Line
Guidance
In the previous concept, we found the slope between two points. We will now find the entire equation of a line. Recall from Algebra I that the equation of a line in slope-intercept form is \begin{align*}y = mx + b,\end{align*} where \begin{align*}m\end{align*} is the slope and \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}intercept. You can find the slope either by using slope triangles or the Slope Formula. To find the \begin{align*}y-\end{align*}intercept, or \begin{align*}b,\end{align*} you can either locate where the line crosses the \begin{align*}y-\end{align*}axis (if given the graph) or by using algebra.
Example A
Find the equation of the line below.
Solution: Analyze the line. We are given two points on the line, one of which is the \begin{align*}y-\end{align*}intercept. From the graph, it looks like the line passes through the \begin{align*}y-\end{align*}axis at (0, 4), making \begin{align*}b = 4\end{align*}. Now, we need to find the slope. You can use slope triangles or the Slope Formula. Using slope triangles, we have:
From this, we see that the slope is \begin{align*}- \frac{2}{6}\end{align*} or \begin{align*}- \frac{1}{3}\end{align*}.
Plugging our found information into the slope-intercept equation, the equation of this line is \begin{align*}y = - \frac{1}{3} x + 4\end{align*}.
Alternate Method: If we had used the Slope Formula, we would use (0, 4) and (6, 2), which are the values of the given points.
\begin{align*}m = \frac{2-4}{6-0} = \frac{-2}{6} = - \frac{1}{3}\end{align*}
Example B
The slope of a line is -4 and the \begin{align*}y-\end{align*}intercept is (0, 3). What is the equation of the line?
Solution: This problem explicitly tells us the slope and \begin{align*}y-\end{align*}intercept. The slope is -4, meaning \begin{align*}m = -4\end{align*}. The \begin{align*}y-\end{align*}intercept is (0, 3), meaning \begin{align*}b = 3\end{align*}. Therefore, the equation of the line is \begin{align*}y = -4x + 3\end{align*}.
Example C
The slope of a line is \begin{align*}\frac{1}{2}\end{align*} and it passes through the point (4, -7). What is the equation of the line?
Solution: In this problem, we are given \begin{align*}m\end{align*} and a point on the line. The point, (4, -7) can be substituted in for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in the equation. We need to solve for the \begin{align*}y-\end{align*}intercept, or \begin{align*}b\end{align*}. Plug in what you know to the slope-intercept equation.
\begin{align*}y &= mx + b\\ -7 &=\frac{1}{2}(4) + b\\ -7 &=2 + b\\ -9 &=b\end{align*}
From this, the equation of the line is \begin{align*}y = \frac{1}{2}x - 9\end{align*}.
We can test if a point is on a line or not by plugging it into the equation. If the equation holds true, the point is on the line. If not, then the point is not on the line.
Example D
Find the equation of the line that passes through (12, 7) and (10, -1).
Solution: In this example, we are not given the slope or the \begin{align*}y-\end{align*}intercept. First, we need to find the slope using the Slope Formula.
\begin{align*}m = \frac{-1-7}{10-12} = \frac{-8}{-2} = 4\end{align*}
Now, plug in one of the points for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. It does not matter which point you choose because they are both on the line.
\begin{align*}7 &= 4(12) + b\\ 7 &= 48 + b \\ -41 &= b\end{align*}
The equation of the line is \begin{align*}y = 4x - 41\end{align*}.
Intro Problem Revisit To determine the equation of the line, rewrite the given information as points. The first could be (0, 800) and the second would be (3, 450). We already know that the y-intercept is 800 because the x-value is zero at that point. Find the slope.
\begin{align*}\frac{800-450}{0-3}= - \frac{350}{3}\end{align*}
Therefore, the equation of the declining value of the laptop is \begin{align*}y= - \frac{350}{3}x + 800\end{align*}. In 6 years, the laptop will be worth \begin{align*}y= - \frac{350}{3}\cdot 6 + 800 = -700+800 = 100\end{align*}. The laptop will be worth $100.
Guided Practice
1. What is the equation of the line where the slope is 1 and passes through (5, 3)?
2. Find the equation of the line that passes through (9, -4) and (-1, -8).
3. Find the equation of the line below.
Answers
1. We are told that \begin{align*}m = 1, x = 5,\end{align*} and \begin{align*}y = 3\end{align*}. Plug this into the slope-intercept equation and solve for \begin{align*}b\end{align*}.
\begin{align*}3 &= 1(5) + b\\ 3 &= 5 + b \\ -2 &= b\end{align*}
The equation of the line is \begin{align*}y = x - 2\end{align*}
2. First, find the slope.
\begin{align*}m = \frac{-8-(-4)}{-1-9} = \frac{-4}{-10} = \frac{2}{5}\end{align*}
Now, find the \begin{align*}y-\end{align*}intercept. We will use the second point. Remember, it does not matter which point you use.
\begin{align*}-8 &= \frac{2}{5}(-1) + b\\ -8 &= - \frac{2}{5} + b \\ -7 \frac{3}{5} &= b\end{align*}
The equation of the line is \begin{align*}y = \frac{2}{5}x - 7 \frac{3}{5}\end{align*} or \begin{align*}y = \frac{2}{5}x - \frac{38}{5}\end{align*}.
When your \begin{align*}y-\end{align*}intercept is a fraction, make sure it is reduced. Double-check with your teacher on how s/he wants you to leave your answer.
3. We can find the slope one of two ways: using slope triangles or by using the Slope Formula. We are given (by the drawn points in the picture) that (-2, 2) and (4, -2) are on the line. Drawing a slope triangle, we have:
We have that the slope is \begin{align*}- \frac{4}{6}\end{align*} or \begin{align*}- \frac{2}{3}\end{align*}. To find the \begin{align*}y-\end{align*}intercept, it looks like it is somewhere between 0 and 1. Take one of the points and plug in what you know to the slope-intercept equation.
\begin{align*}2 &= - \frac{2}{3}(-2) + b\\ 2 &= \frac{4}{3} + b \\ \frac{2}{3} &= b\end{align*}
The equation of the line is \begin{align*}y = - \frac{2}{3}x + \frac{2}{3}\end{align*}.
Vocabulary
- Slope-Intercept Form
- The equation of a line in the form \begin{align*}y = mx + b,\end{align*} where \begin{align*}m\end{align*} is the slope and \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}intercept.
- \begin{align*}y-\end{align*}intercept
- The point where a line crosses the \begin{align*}y-\end{align*}axis. This point will always have the form \begin{align*}(0, y)\end{align*}.
- \begin{align*}x-\end{align*}intercept
- The point where a line crosses the \begin{align*}x-\end{align*}axis. This point will always have the form \begin{align*}(x, 0)\end{align*}.
Practice
Find the equation of each line with the given information below.
- slope = 2, \begin{align*}y-\end{align*}intercept = (0, 3)
- \begin{align*}m = -\frac{1}{4}, \ b = 2.6\end{align*}
- slope = -1, \begin{align*}y-\end{align*}intercept = (0, 2)
- \begin{align*}x-\end{align*}intercept = (-2, 0), \begin{align*}y-\end{align*}intercept = (0, -5)
- slope \begin{align*}= \frac{2}{3}\end{align*} and passes through (6, -4)
- slope \begin{align*}= - \frac{3}{4}\end{align*} and passes through (-2, 5)
- slope = -3 and passes through (-1, -7)
- slope = 1 and passes through (2, 4)
- passes through (-5, 4) and (1, 1)
- passes through (5, -1) and (-10, -10)
- passes through (-3, 8) and (6, 5)
- passes through (-4, -21) and (2, 9)
For problems 13-16, find the equation of the lines in the graph below.
- Green Line
- Blue Line
- Red Line
- Purple Line
- Find the equation of the line with zero slope and passes through (8, -3).
- Find the equation of the line with zero slope and passes through the point (-4, 5).
- Find the equation of the line with zero slope and passes through the point \begin{align*}(a, b)\end{align*}.
- Challenge Find the equation of the line with an undefined slope that passes through \begin{align*}(a, b)\end{align*}.
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Color | Highlighted Text | Notes | |
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Term | Definition |
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intercept | An intercept is a location where a graph crosses the axis. As a coordinate pair, this point will always have the form . intercepts are also called solutions, roots or zeros. |
intercept | A intercept is a location where a graph crosses the axis. As a coordinate pair, this point will always have the form . |
Slope-Intercept Form | The slope-intercept form of a line is where is the slope and is the intercept. |