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### Course: Operations and Algebraic Thinking 229+ > Unit 3

Lesson 1: Intro to slope-intercept form# Intro to slope-intercept form

Learn about the slope-intercept form of two-variable linear equations, and how to interpret it to find the slope and y-intercept of their line.

#### What you should be familiar with before taking this lesson

- You should know what
*two-variable linear equations*are. Specifically, you should know that the graph of such equations is a line. If this is new to you, check out our intro to two-variable equations. - You should also be familiar with the following properties of linear equations:
-intercept and$y$ -intercept and slope.$x$

#### What you will learn in this lesson

- What is the
**slope-intercept form**of two-variable linear equations - How to find the slope and the
-intercept of a line from its slope-intercept equation$y$ - How to find the equation of a line given its slope and
-intercept$y$

## What is slope-intercept form?

*Slope-intercept*is a specific form of linear equations. It has the following general structure. Drum roll ...

Here, ${m}$ and ${b}$ can be any two real numbers. For example, these are linear equations in slope-intercept form:

$y=2x+1$ $y=-3x+2.7$ $y=10-100x$

On the other hand, these linear equations are

*not*in slope-intercept form:$2x+3y=5$ $y-3=2(x-1)$ $x=4y-7$

Slope-intercept is the most prominent form of linear equations. Let's dig deeper to learn why this is so.

## The coefficients in slope-intercept form

Besides being neat and simplified, slope-intercept form's advantage is that it gives two main features of the line it represents:

- The slope is
.${m}$ - The
-coordinate of the$y$ -intercept is$y$ . In other words, the line's${b}$ -intercept is at$y$ .$(0,{b})$

For example, the line $y={2}x{+1}$ has a slope of ${2}$ and a $y$ -intercept at $(0,{1})$ :

The fact that this form gives the slope and the $y$ -intercept is the reason why it is called

*slope-intercept*in the first place!## Check your understanding

## Why does this work?

You might be wondering how it is that in slope-intercept form, ${m}$ gives the slope and ${b}$ gives the $y$ -intercept.

Can this be some sort of magic? Well, it certainly is $y={2}x+{1}$ as an example.

*not*magic. In math, there's always a justification. In this section we'll take a look at this property using the equation### Why ${b}$ gives the $y$ -intercept

At the $y$ -intercept, the $x$ -value is always zero. So if we want to find the $y$ -intercept of $y={2}x+{1}$ , we should substitute $x=0$ and solve for $y$ .

We see that at the $y$ -intercept, ${2}x$ becomes zero, and therefore we are left with $y={1}$ .

### Why ${m}$ gives the slope

Let's refresh our memories about what slope is exactly. Slope is the ratio of the change in $y$ over the change in $x$ between any two points on the line.

If we take two points where the change in $x$ is exactly $1$ unit, then the change in $y$ will be equal to the slope itself.

Now let's look at what happens to the $y$ -values in the equation $y={2}x+{1}$ as the $x$ -values constantly increase by $1$ unit.

We see that each time $x$ increases by $1$ unit, $y$ increases by ${2}$ units. This is because $x$ determines the multiple of ${2}$ in the calculation of $y$ .

As stated above, the change in $y$ that corresponds to $x$ increasing by $1$ unit is equal to the slope of the line. For this reason, the slope is ${2}$ .

## Want to join the conversation?

- What if m=0?(59 votes)
- If the slope is 0, is a horizontal line. It makes sense if you think about it. Each time we increase one x, increase y by 0.(154 votes)

- how do you find the slope and intercept on a graph?(32 votes)
- To find the y-intercept, find where the line hits the y-axis. To find the x-intercept (which wasn't mentioned in the text), find where the line hits the x-axis. To find the slope, find two points on the line then do y2-y1/x2-x1
*the numbers are subscripts*.

Hope that helped.(46 votes)

- I dont understand this whole thing at all PLEASE HELP!(26 votes)
- The slope-intercept form of a linear equation is where one side contains just "y". So, it will look like: y = mx + b where "m" and "b" are numbers.

This form of the equation is very useful. The coefficient of "x" (the "m" value) is the slope of the line. And, the constant (the "b" value) is the y-intercept at (0, b)

So, if you are given an equation like: y = 2/3 (x) -5

We can tell that the slope of the line = 2/3 and the y-intercept is at (0, -5)

Hope this helps.(39 votes)

- how does an equation result to an answer?(24 votes)
- The equation results in how to graph the line on a graph. If they give you the x value then you would plug that in and it would tell you the answer in y.(11 votes)

- Why should I learn this and what can I use this for in the future.(16 votes)
- slopes are all over the place in the real world, so it depends on what you plan to do in life of how much you use this. Art, building, science, engineering, finance, statistics, etc. all use linear functions.(18 votes)

- Is it ever possible that the slope of a linear function can fluctuate? Or is the slope always a fixed value?(10 votes)
- A linear functions creates a straight line. So, the slope is
**always**a fixed value. If the slope could fluctuate, then you wouldn't have a straight line graph. The graph would be curved, or V-shaped.(14 votes)

- Why is it called algebra? Is it Greek or something?(7 votes)
- There is an overview history video in Algebra 1 that explains this better than I can but basically Algebra is a Medieval Latin short hand for the title of the first book explaining these principals.

It was called "al-mukhtasar fi hisab al-jabr wa al-muqabala", which is Arabic for "the compendium on calculation by restoring and balancing".

Here's the link to the vid if you want to explore further: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:foundation-algebra/x2f8bb11595b61c86:algebra-overview-history/v/origins-of-algebra

Hope I answered your question well! Best of luck learning🍀(24 votes)

- say you have a problem like (3,1) slope= 4/3. how would you work that out(3 votes)
- Pretty late here, but for anyone else reading, I'll assume they meant how you find the slope intercept using only these values.

Since we know the slope is 4/3, we can conclude that: y = 4/3 * x ... But what is the constant, the y axis intercept point?

You can solve for it by doing: 1 = 4/3 * 3 + c... We know the values for x and y at some point in the line, but we want to know the constant, c. You can solve this algebraically.

1 = 4/3 * 3 + c

1 = 4 + c

1 - 4 = 4 - 4 + c

-3 = c

The slope intercept equation is: y = 4/3 * x - 3

The y axis intercept point is: (0 , -3)

I just started learning this so if anyone happens across this and spots an error lemme know.(8 votes)

- I am so lost I need help :(((6 votes)
- The start of the lesson states what you should have some understanding of, so the first question is do you have some understanding of these two concepts?

"You should know what two-variable linear equations are. Specifically, you should know that the graph of such equations is a line. If this is new to you, check out our intro to two-variable equations.

You should also be familiar with the following properties of linear equations: y-intercept and x-intercept and slope."

If these are an issue, you need to go back and review these concepts. If you understand these, then you need to be more specific on where you are struggling. This is just an intro, so it is basically identifying slope and intercept from an equation. Which checks do not make sense?(4 votes)

- How do I find the x intercept?(3 votes)
- You use y=0 in the equation and calculate "x"(8 votes)