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Coordinate plane Plot points on a graph Linear graphHere you will learn about the slope-intercept form of a line, y=m x+b, including how to recognize the slope and y -intercept from the equation y=m x+b and rearrange an equation into slope intercept form y=m x+b.

Students first work with linear equations in 7 th grade when they learn about the constant of proportionality and direct variation. They expand their knowledge as they move through 8 th grade and high school math.

y=m x+b is the slope intercept form of the line. There are other ways to write the equation of a straight line like standard form or point slope form. However, the focus here is going to be on the slope-intercept form of the equation of a line, y=m x+b.

- m represents the slope of a line which is how steep the line is
- b represents the y -intercept of the line which is the point where the line intersects the y -axis
- x value represents the x -coordinate of any point on the line
- y value represents the y -coordinate of any point on the line

y=m x+b is a linear equation because when it’s graphed on the coordinate plane, it forms a line. You can also determine this because the power of x is equal to 1.

Let’s take a look at the linear equation y=2 x+1.

Since m is represented by the number 2 and b is represented by the number 1. You can state that the slope of this line is 2 and the y -intercept is 1.

Graphically, you can see that the ordered pair of the y -intercept is (0, 1) and the slope is represented by 2 units up and 1 unit to the right.

Since the x and y represent the coordinates of any point on the line, you can create a table of values to find ordered pairs to represent the line graphically. Recall that the independent variable is x and the dependent variable is y.

Here is a quick summary of some equations in the form y=m x+b with the slope and y intercept highlighted.

**Special cases:**

*Horizontal lines*

Horizontal lines have a slope of 0.

So, the equation in slope-intercept form of the horizontal line below can be expressed as, y=0 x+3 which can be written as y=3.

*Vertical lines:*

Vertical lines have an undefined slope.

So, the equation of the line graphed below can be expressed as x=3. The equation of all vertical lines will be the x value of where the line intersects the x axis.

How does this relate to 8 th grade math and high school math?

**Grade 8 Functions (8.F.A.3)**Interpret the equation y=m x+b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A=s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2, 4) and (3,9), which are not on a straight line.

**Grade 8 Expressions and Equations (8.EE.B.6)**

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = m x for a line through the origin and the equation y = m x + b for a line intercepting the vertical axis at b.

**High School Functions: Interpreting Functions (HSF-IF.C.7a)**

Graph linear and quadratic functions and show intercepts, maxima, and minima.

**High School Functions: Linear, Quadratic, and Exponential Models (HSF-LE.A.2)**

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Use this quiz to check your grade 6 – grade 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th to 8th grade algebra topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 6 – grade 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th to 8th grade algebra topics to identify areas of strength and support!

DOWNLOAD FREEIn order to state the slope and y-intercept of a line:

**Rearrange the equation to make sure it is in the form of \textbf{y = mx + b}.****Identify the numbers that represent \textbf{m} and \textbf{b}.****State the slope as rise over run and the \textbf{y} -intercept as an ordered pair.**

State the slope and y -intercept of the line y=-3 x+8.

**Rearrange the equation to make sure it is in the form of \textbf{y = mx + b}.**

The equation y=-3 x+8 is already in slope intercept form, y=m x+b, so you can progress to step 2.

2**Identify the numbers that represent \textbf{m} and \textbf{b}. **

Comparing the given equation of y=-3 x+8 to the slope intercept form of a linear equation, y=m x+b.

You can conclude that:

m=-3 because it is the coefficient of the x term

b=8 because it is the constant

3**State the slope as rise over run and the \textbf{y} -intercept as an ordered pair.**

Since the slope is -3 which is the same as -\cfrac{3}{1}, the slope of this line is 3 units down and 1 unit to the right OR 3 units up and 1 unit left.

The y -intercept is 8 which means the line crosses the y -axis at 8 so the coordinate of the y -intercept is (0, 8). You can also find this by plugging in a 0 for x and solving for y.

\begin{aligned}&y=-3 x+8 \\\\ &\begin{aligned}& y=-3(0)+8 \\\\ & y=0+8 \\\\ & y=8 \end{aligned}\end{aligned}Find the slope and y -intercept of the line y=7-x.

**Rearrange the equation to make sure it is in the form of \textbf{y = mx + b}. **

The given equation is not in the form of y=m x+b.

By rearranging the terms, you can write it in the correct form.

y=7-x

y=-x+7 (simply rearrange the right handside of the equation writing the x term first and then the constant term.)

**Identify the numbers that represent \textbf{m} and \textbf{b}. **

Comparing the rearranged equation of y=-x+7 to the slope intercept form of a linear equation, y=m x+b

You can conclude that:

m=-1 because it is the coefficient of the x term

b=7 because it is the constant term

**State the slope as rise over run and the \textbf{y} -intercept as an ordered pair.**

Since the slope is -1 which is the same as -\cfrac{1}{1}, the slope of this line is 1 unit down and 1 unit to the right OR 1 unit up and 1 unit left.

The y -intercept is 7 which means the line crosses the y -axis at 7 so the coordinate of the y -intercept is (0, 7). You can also find this by plugging in a 0 for x and solving for y.

\begin{aligned}&\begin{aligned}& y=-x+7 \\\\
& y=-1(0)+7 \\\\
& y=0+7\end{aligned}\\\\
&y=7\end{aligned}

Find the slope and y -intercept of the line x=y+10.

**Rearrange the equation to make sure it is in the form of \textbf{y = mx + b}. **

The given equation is not in the slope intercept form equation, y=m x+b. So, you need to solve the equation, x=y+10, for y.

\begin{aligned}& x=y+10 \\\\
& x-y=y-y+10 \\\\
& x-y=10 \\\\
& x-x-y=10 \\\\
& -y=-x+10 \\\\
& \cfrac{-y}{-1}=\cfrac{-x}{-1}+\cfrac{10}{-1} \\\\
& y=x-10\end{aligned}

**Identify the numbers that represent \textbf{m} and \textbf{b}. **

Comparing the rearranged equation of y=x-10 to the slope intercept form of a linear equation, y=m x+b

You can conclude that:

m=1 because it is the coefficient of the x term

b=-10 because it is the constant term

**State the slope as rise over run and the \textbf{y} -intercept as an ordered pair.**

Since the slope is 1 which is the same as \cfrac{1}{1}, the slope of this line is 1 unit down and 1 unit to the right.

The y -intercept is -10 which means the line crosses the y -axis at -10 so the coordinate of the y -intercept is (0, -10). You can also find this by plugging in a 0 for x and solving for y.

\begin{aligned}& y=x-10 \\\\
& y=1(0)-10 \\\\
& y=0-10 \\\\
& y=-10\end{aligned}

Find the slope and y -intercept of the line 2 x=6 y-15.

**Rearrange the equation to make sure it is in the form of \textbf{y = mx + b}. **

The given equation is not in the form of y=m x+b. So, you need to solve the equation, 2 x=6 y-15, for y.

\begin{aligned}&\begin{aligned}& 2 x=6 y-15 \\\\ & 2 x-6 y=6 y-6 y-15 \\\\ & 2 x-2 x-6 y=-2 x-15 \\\\ & -6 y=-2 x-15\end{aligned}\\\\ &\cfrac{-6 y}{-6}=\cfrac{-2 x}{-6}-\cfrac{15}{-6}\end{aligned}

y=\cfrac{1 x}{3}+\cfrac{15}{6} \;\, (simplify the fractions when necessary)

y=\cfrac{1 x}{3}+\cfrac{5}{2}

Remember that,

\begin{aligned}& \cfrac{1 x}{3}=\cfrac{1}{3} x \\\\ & \cfrac{5}{2}=2 \cfrac{1}{2}\end{aligned}

So you can write the equation as, y=\cfrac{1}{3} x+2 \cfrac{1}{2}.

**Identify the numbers that represent \textbf{m} and \textbf{b}. **

Comparing the rearranged equation of y=\cfrac{1}{3} x+2 \cfrac{1}{2} to the slope intercept form of a linear equation, y=m x+b.

You can conclude that:

m=\cfrac{1}{3} because it is the coefficient of the x term

b=2 \cfrac{1}{2} because it is the constant term

**State the slope as rise over run and the \textbf{y} -intercept as an ordered pair.**

Since the slope of this line is \cfrac{1}{3}, this means that the slope is 1 unit up and 3 units right.

The y -intercept is 2 \cfrac{1}{2} which means the line crosses the y -axis at 2 \cfrac{1}{2} so the coordinate of the y -intercept is (0, 2 \cfrac{1}{2} \, ).

You can also find this by plugging in a 0 for x and solving for y.

\begin{aligned}&y=\cfrac{1}{3}\, x+2 \cfrac{1}{2}\\\\ &\begin{aligned}& y=\cfrac{1}{3}\,(0)+2 \cfrac{1}{2} \\\\ &y=0+2 \cfrac{1}{2} \\\\ & y=2 \cfrac{1}{2}\end{aligned}\end{aligned}

Find the slope and y -intercept of the line 3 x=4(y-5) .

**Rearrange the equation to make sure it is in the form of \textbf{y = mx + b}. **

The given equation is not in the form of y=m x+b. So, you need to solve the equation, 3 x=4(y-5), for y.

\begin{aligned}&\begin{aligned}& 3 x=4(y-5) \\\\ & 3 x=4 y-20 \\\\ & 3 x-4 y=4 y-4 y-20 \\\\ & 3 x-3 x-4 y=-3 x-20 \\\\ & -4 y=-3 x-20\end{aligned}\\\\ &\cfrac{-4 y}{-4}=\cfrac{-3 x}{-4}-\cfrac{20}{-4}\end{aligned}

y=\cfrac{3 x}{4}+\cfrac{20}{4} \;\, (simplify the fractions when necessary)

y=\cfrac{3 x}{4}+5

Remember that,

\cfrac{3 x}{4}=\cfrac{3}{4} x

So you can write the equation as, y=\cfrac{3}{4} x+5

**Identify the numbers that represent \textbf{m} and \textbf{b}. **

Comparing the rearranged equation of y=\cfrac{3}{4} x+5 to the slope intercept form of a linear equation, y=m x+b.

You can conclude that:

m=\cfrac{3}{4} because it is the coefficient of the x term

b=5 because it is the constant term

**State the slope as rise over run and the \textbf{y} -intercept as an ordered pair.**

Since the slope of this line is \cfrac{3}{4}, this means that the slope is 3 units up and 4 units right.

The y -intercept is 5 which means the line crosses the y -axis at 5 so the coordinate of the y -intercept is (0, 5). You can also find this by plugging in a 0 for x and solving for y.

\begin{aligned}& y=\cfrac{3}{4} x+5 \\\\
& y=\cfrac{3}{4}(0)+5 \\\\
& y=0+5\end{aligned}

Find the slope and y -intercept of the line x=\cfrac{y+0.85}{0.2}.

**Rearrange the equation to make sure it is in the form of \textbf{y = mx + b}. **

The given equation is not in the form of y=m x+b. So, you need to solve the equation, x=\cfrac{y+0.85}{0.2}, for y.

\begin{aligned}& x=\cfrac{y+0.85}{0.2} \\\\ & x \times 0.2=\cfrac{y+0.85}{0.2} \times 0.2 \\\\ & 0.2 x=y+0.85 \\\\ & 0.2 x-y=y-y+0.85 \\\\ & 0.2 x-0.2 x-y=-0.2 x+0.85 \\\\ & -y=-0.2 x+0.85 \\\\ & \cfrac{-y}{-1}=\cfrac{-0.2 x}{-1}+\cfrac{0.85}{-1} \\\\ & y=0.2 x-0.85\end{aligned}

Remember that,

\begin{aligned}& 0.2=\cfrac{2}{10}=\cfrac{1}{5} \\\\ & 0.85=\cfrac{85}{100}=\cfrac{17}{20}\end{aligned}

So you can write the equation as, y=0.2 x+0.85 OR y=\cfrac{1}{5} x+\cfrac{17}{20}.

**Identify the numbers that represent \textbf{m} and \textbf{b}. **

Comparing the rearranged equation of y=0.2 x+0.85, which is the same as y=\cfrac{1}{5} x+\cfrac{17}{20}, to the slope intercept form of a linear equation, y=m x+b.

You can conclude that:

m=0.2 or \cfrac{1}{5} because it is the coefficient of the x term

b=0.85 or \cfrac{17}{20} because it is the constant term

*To interpret the slope, it is best to have a fraction.*

**State the slope as rise over run and the \textbf{y} -intercept as an ordered pair.**

Since the slope of this line is \cfrac{1}{5}, this means that the slope is 1 unit up and 5 units right.

The y -intercept is 0.85 or \cfrac{17}{20} which means the line crosses the y -axis at 0.85 or \cfrac{17}{20} so the coordinate of the y -intercept is (0, 0.85) or (0, \cfrac{17}{20}\,).

You can also find this by plugging in a 0 for x and solving for y.

\begin{aligned}& y=0.2 x+0.85 \\\\
& y=0.2(0)+0.85 \\\\
& y=0+0.85 \\\\
& y=0.85\end{aligned}

- Use digital graphing platforms such as
*Desmos*so that students can connect the algebraic equation to the visual representation of the graph.

- Expose students using
*Desmos*to special cases, such as when the y -intercept is 0 or when the slope is 0.

- When students are first learning the concept, encourage them to write the equation in correct slope intercept form where the x term is first and then the constant.

- Always provide creative opportunities for students such as having them create designs using linear equations on
*Desmos*.

- Instead of a worksheet, have students practice skills using digital platforms like Khan Academy.

**Using the incorrect inverse operation**

When rearranging equations, instead of applying the inverse operation to the term being moved, the value is simply moved to the other side of the equals sign.

For example,

y+5=2 x is rearranged to make y=2 x+5.

**Stating the value of \textbf{m} and \textbf{b} when the equation is not in the form \textbf{y = mx + b}**

For example, in the equation, x=y+10 stating that the m or slope is 1 and that the b or y -intercept is 10. Instead of first rearranging the equation to be in slope intercept form.

x=y+10

x-10=y+10-10

x-10=y which is the same as y=x-10

m=1 (slope)

b=-10 ( y -intercept)

**Mixing up the slope and the \textbf{y} -intercept**

For example, the equation y=10+3 x. Since the equation is not quite in slope-intercept form, y=m x+b, students might think that the slope is 10 because it is written first and that the y -intercept is 3. The coefficient of x is 3 which means the slope is 3.

Also, when x = 0, \, y=10+3(0), the value of y is 10 which means the y -intercept is 10. Students can also rearrange the equation to be in slope intercept form to avoid confusion, y=3 x+10.

1. Find the slope, m, and y -intercept, b, for the given equation.

y=-5 x+9

m=-5, \; b=9

m=5, \; b = 9

m=9, \; b =5

m=9, \; b=-5

The slope intercept form of a linear equation is y=m x+b.

In this case, y=-5 x+9, \, m=-5 and b=9.

The coefficient of the x term is -5 so it is the slope.

If you make x=0 you can solve for the y -intercept.

\begin{aligned}& y=-5(0)+9 \\\\ & y=9\end{aligned}

So the y -intercept is 9.

2. Find the slope, m, and y -intercept, b, for the given equation.

y=6-x

m=6, \; b=-1

m=1, \; b=6

m=-1, \; b=6

m=0, \; b=6

The slope intercept form of a linear equation is y=m x+b.

In this case, y=6-x, which should be rewritten in the form of y=m x+b.

You can switch the terms and write the equation as y=-x+6 where m=-1 and b=6.

The coefficient of the x term is -1 so it is the slope.

If you make x=0 you can solve for the y -intercept.

\begin{aligned}& y=-1(0)+6 \\\\ & y=6\end{aligned}

So the y -intercept is 6.

b=6

3. Find the slope, m, and y -intercept, b, for the given equation.

x=2 y+5

m=2, \; b=5

m=\cfrac{1}{2}, \; b=-\cfrac{5}{2}

m=\cfrac{1}{2}, \; b=5

m=1, \; b=5

First put the equation in y=m x+b form first which means to solve the equation for y.

\begin{aligned}& x=2 y+5 \\\\ & x-5=2 y+5-5 \\\\ & x-5=2 y \\\\ & \cfrac{x}{2}-\cfrac{5}{2}=\cfrac{2 y}{2} \end{aligned}

\cfrac{x}{2}-\cfrac{5}{2}=y which is the same as y=\cfrac{x}{2}-\cfrac{5}{2}

Remember that \cfrac{x}{2}=\cfrac{1}{2} \, x.

So the slope = m=\cfrac{1}{2} and the y -intercept = b=\cfrac{5}{2}=2 \cfrac{1}{2} \, .

4. Find the equation of a line in y=m x+b form that has a slope of -5 and a y -intercept of 0?

y=0 x-5

y=5 x-0

y=-5 x

y=x-5

The slope is given to be -5 and the y -intercept is given to be 0.

So, m=-5 and b=0.

Using those values, substitute them into y=m x+b, which gives the equation y=-5 x+0. That is the same as y=-5 x.

5. What is the slope and the y -intercept of the equation below?

2 x=3(3+y)

m=\cfrac{2}{3}, \; b=-3

m=2, \; b=0

m=6, \; b=9

m=\cfrac{2}{3}, \; b=3

First put the equation in y=m x+b form first which means to solve the equation for y.

\begin{aligned}& 2 x=3(3+y) \\\\ & 2 x=9+3 y \\\\ & 2 x-9=9-9+3 y \\\\ & 2 x-9=3 y \\\\ & \cfrac{2 x}{3}-\cfrac{9}{3}=\cfrac{3 y}{3} \end{aligned}

\cfrac{2 x}{3}-3=y which is the same as y=\cfrac{2 x}{3}-3

Remember that \cfrac{2 x}{3}=\cfrac{2}{3} \, x.

So the slope = m=\cfrac{2}{3} and the y -intercept = b=-3.

6. What is the slope and the y -intercept of the equation below?

0.5 x+0.75 y=0.25

m=0.75, \; b=0.5

m=-\cfrac{2}{3}, \; b=\cfrac{1}{3}

m=\cfrac{1}{3}, \; b=\cfrac{2}{3}

m=1.5, \; b=0.5

First put the equation in y=m x+b form first which means to solve the equation for y.

\begin{aligned}& 0.5 x+0.75 y=0.25 \\\\ & 0.5 x-0.5 x+0.75 y=-0.5 x+0.25 \\\\ & 0.75 y=-0.5 x+0.25 \\\\ & \cfrac{0.75 y}{0.75}=\cfrac{-0.5 x}{0.75}+\cfrac{0.25}{0.75} \\\\ & y=-0 . \overline{6} x+0 . \overline{3}\end{aligned}

Recall:

\begin{aligned}& -0 . \overline{6}=\cfrac{6}{9}=\cfrac{2}{3} \\\\ & 0 . \overline{3}=\cfrac{3}{9}=\cfrac{1}{3} \\\\ & \text { So, } y=-\cfrac{2}{3} \, x+\cfrac{1}{3}\end{aligned}

The slope = m=-\cfrac{2}{3} (negative slope) and the y -intercept = b=\cfrac{1}{3}

The x -intercept is the point where the line passes or crosses the x -axis.

Yes, you can use the slope formula to find the slope of a line using two points on the line.

To calculate the slope, subtract the y -coordinates and the x -coordinates, and write them as a quotient of the differences of the y’s over the differences of the x’s.

The slope intercept equation is y=m x+b, where m represents the slope and b represents the y -intercept.

- Angles
- Rate of change
- Systems of equations
- Number patterns

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