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Determining the Equation of a Line

Write equations from y-intercept and slope or two points

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Practice Determining the Equation of a Line
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Writing Linear Equations

Have you ever studied banana plants? Take a look at this dilemma.

Mr. Thomas' class is having a discussion on the rainforest. The students have been doing some research, and now they are ready to talk about their findings.

“The vegetation of the rainforest was very interesting,” Carmen commented in Mr. Thomas’ class.

“There were so many different things growing,” Mark agreed.

The students began having a discussion about the things that had intrigued them about the plant life of the rainforest. One of the points that was brought up is that there are plants in the rainforest that can’t be found anywhere else in the world. Mr. Thomas spotted the opportunity and wrote the following problem.

You buy a banana tree that is 8 inches tall. It grows 4 inches per day. Its height (in inches) h\begin{align*}h\end{align*} is a function of time (in days) d\begin{align*}d\end{align*}.

You can express this function as an equation. This Concept will show you how to write linear equations.

Guidance

The y=mx+b\begin{align*}y=mx+b\end{align*} form of an equation was most useful in rapidly identifying both the slope, m\begin{align*}m\end{align*}, and the y\begin{align*}y\end{align*}-intercept, b\begin{align*}b\end{align*}. In fact, if we know what the slope of an equation is and we know its y\begin{align*}y\end{align*}-intercept, then we can just as easily write the equation. All you have to do is plug in the slope for m\begin{align*}m\end{align*} in the y=mx+b\begin{align*}y=mx+b\end{align*} form and the y\begin{align*}y\end{align*}-intercept for b\begin{align*}b\end{align*}.

Take a look at this situation.

m=4,yintercept=3\begin{align*}m = 4, y-intercept = 3\end{align*}

We know that we are going to use the form of the equation y=mx+b\begin{align*}y=mx+b\end{align*}, so we can substitute these values into the equation and write it.

y=4x+3\begin{align*}y=4x+3\end{align*}

This is the answer. The key is to always watch for negative signs and be sure to include them when you write your equation.

Sometimes, we can also be given the slope and a point that the line crosses through. We can also use this information to write an equation of a line.

Slope=2\begin{align*}\text{Slope} = -2\end{align*}, the line passes through the point (0,-3)

With this example, we know the slope, so that can be easily substituted into the slope-intercept form. The point has a 0 for the x\begin{align*}x\end{align*} value, so that means that we have been given the coordinate of the y\begin{align*}y\end{align*} – intercept.

y=2x3\begin{align*}y= -2x-3\end{align*}

This is the answer.

What if you only know two points and you don’t know the slope? It’s a similar operation that we can use in order to write the equation.

Do you recall that the slope formula is m=y2y1x2x1\begin{align*}m=\frac{{y_2}-{y_1}}{{x_2}-{x_1}}\end{align*}? In other words, given any two points (x1,y1)\begin{align*}(x_1, y_1)\end{align*} and (x2,y2)\begin{align*}(x_2, y_2)\end{align*}, we can use the slope formula to calculate the slope of the line that passes through those points. Even when you only know two points, finding the slope is just a matter of using the formula. But then what?

We will use the notion below.

if m=y2y1x2x1\begin{align*}m=\frac{{y_2}-{y_1}}{{x_2}-{x_1}}\end{align*} then m=yy1xx1\begin{align*}m=\frac{{y}-{y_1}}{{x}-{x_1}}\end{align*} because the slope is the same on any part of a line. In other words, we can use a formula similar to the slope formula for finding the equation. This time, however, we will leave x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} as variables because the relationship is true for any values of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} in that equation.

Write the equation of the line that passes through the points (3,7)\begin{align*}(3,7)\end{align*} and (5,11)\begin{align*}(5,11)\end{align*}. First we will find the slope using the slope formula for x1=3,y1=7,x2=5,y2=11\begin{align*}{x_1}=3,{y_1}=7,{x_2}=5,{y_2}=11\end{align*}.

mmmm=y2y1x2x1=11753=42=2

Now plug in our known values of m,x1\begin{align*}m, x_1\end{align*}, and y1\begin{align*}y_1\end{align*}.

m21=yy1xx1=y3x7

Do you see that we have a proportion? This can be solved by cross multiplying.

211(y3)y3y=y3x7=2(x7)=2x14=2x11

This is the answer.

What about when you have been given a table of values? Well, there is a way to figure out the equation quite simply when you have a table of values. Take a look.

x\begin{align*}x\end{align*} y\begin{align*}y\end{align*}
0 5
1 7
2 9
3 11
4 13

First, notice that the y\begin{align*}y\end{align*} – intercept is the value that has an x\begin{align*}x\end{align*} value of 0. With an x\begin{align*}x\end{align*} value of 0, we know that the y\begin{align*}y\end{align*} – intercept is 5.

Now we need to figure out the slope. Look at the y values in the table. Can you see a pattern? If you look carefully, you will see that the values jump by +2 each time. This is the slope. Think about how the line would move when graphed. The pattern of the y\begin{align*}y\end{align*} values represents the slope of the line.

y=2x+5\begin{align*}y=2x+5\end{align*}

This is the answer.

What about function notation? Well first, think about independent and dependent variables.

In science, an independent variable is a parameter that is manipulated or chosen by a scientist while the dependent variable is a parameter that is measured. Scientists oftentimes look for a correlation between an independent variable and a dependent variable—they want to know if the dependent variable depends on the independent variable. For example, a scientist might measure the speed at which a car is moving and the force upon impact when the car hits a wall. The scientist can manipulate the speed of the car—she can make the car move slower or faster. She would then measure the force of impact related to the given speed. Then, a conclusion can be drawn about their relatedness and cars, in this case, might be designed based on that relation. The independent variable will be shown in the left column of a t-table and on the x\begin{align*}x\end{align*}-axis of a graph. The dependent variable will be shown in the right column of a t-table and on the y\begin{align*}y\end{align*}-axis of a graph.

f(x)=4x+1\begin{align*}f(x)=4x+1\end{align*}

Here we know that the function of x\begin{align*}x\end{align*} is dependent on 4 times that value, x\begin{align*}x\end{align*} and one.

Write a linear equation by using the given information.

Example A

m=2,yintercept=5\begin{align*}m = 2, y-intercept = 5\end{align*}

Solution: y=2x+5\begin{align*}y=2x+5\end{align*}

Example B

\begin{align*}m = -4, y-intercept = 6\end{align*}

Solution: \begin{align*}y=-4x+6\end{align*}

Example C

\begin{align*}m = 8, y-intercept = -2\end{align*}

Solution: \begin{align*}y=8x-2\end{align*}

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

First, we need to write an equation. We can use the \begin{align*}h\end{align*} to represent the height of the banana tree. We can use the \begin{align*}d\end{align*} to represent the number of days. The 8 is the height that the tree started with. Here is our equation.

\begin{align*}h=4d+8\end{align*}

This is our answer.

Vocabulary

Independent Variable
a value that is not dependent on another value. It is the \begin{align*}x\end{align*} value in a table.
Dependent Variable
a value that is dependent on the equation. It is the \begin{align*}y\end{align*} value in a table.
Function Notation
an equation where the value of \begin{align*}x\end{align*} is dependent on the equation involving \begin{align*}x\end{align*}.

Guided Practice

Here is one for you to try on your own.

Write an equation in slope-intercept form with a slope of -4 and a y-intercept of 13.

Solution

To do this, we can take slope-intercept form and substitute in the given values.

\begin{align*}y=mx+b\end{align*}

The \begin{align*}m\end{align*} represents the slope.

The \begin{align*}b\end{align*} represents the y-intercept.

\begin{align*}y=-4x+13\end{align*}

This is our answer.

Practice

Directions: Write the equation of a line with the following slopes and \begin{align*}y\end{align*}intercepts.

1. \begin{align*}\text{slope} = 2, \ y \ \text{int} = 4\end{align*}
2. \begin{align*}\text{slope} = -3, \ y \ \text{int} = 2\end{align*}
3. \begin{align*}\text{slope} = -4, \ y \ \text{int} = 4\end{align*}
4. \begin{align*}\text{slope} = 3, \ y \ \text{int} = -5\end{align*}
5. \begin{align*}\text{slope} = \frac{1}{2}, \ y \ \text{int} = -2\end{align*}
6. \begin{align*}\text{slope} = -\frac{1}{3}, \ y \ \text{int} = 2\end{align*}
7. \begin{align*}\text{slope} = 1, \ y \ \text{int} = 8\end{align*}
8. \begin{align*}\text{slope} = -2, \ y \ \text{int} = 4\end{align*}
9. \begin{align*}\text{slope} = -1, \ y \ \text{int} = -1\end{align*}
10. \begin{align*}\text{slope} = 5, \ y \ \text{int} = -2\end{align*}

Directions: Write the following horizontal or vertical line equations.

1. A horizontal line with a \begin{align*}b\end{align*} value of 7.
2. A horizontal line with a \begin{align*}b\end{align*} value of -4.
3. A vertical line with an \begin{align*}x\end{align*} value of 2
4. A vertical line with an \begin{align*}x\end{align*} value of -5

Directions: Write the equation of a line that passes through the following points.

1. (3, -3) and (-3, 1)
2. (2, 3) and (0, -3)

Vocabulary Language: English

dependent variable

dependent variable

The dependent variable is the output variable in an equation or function, commonly represented by $y$ or $f(x)$.
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 $y$.
independent variable

independent variable

The independent variable is the input variable in an equation or function, commonly represented by $x$.

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