linear equations homework 1 slope

Unit 4 Linear Equations Homework 1 Slope Answer Key

In this article, we will delve into Unit 4 Linear Equations Homework 1 and explore the concept of slope. Slope is a fundamental concept in algebra and plays a crucial role in understanding the relationship between two variables. We will provide a comprehensive answer key to the homework questions, guiding you through the process of finding slopes and interpreting their meanings in real-life scenarios.

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Introduction to Linear Equations and Slope

Linear equations play a fundamental role in algebra and mathematics. They help us understand the relationships between variables and how they change with respect to one another. Among the essential concepts related to linear equations, "slope" stands out as a critical factor. In this article, we will delve into the concept of slope, explore its applications, and provide a comprehensive answer key for Unit 4 Linear Equations Homework 1.

Understanding Slope in Linear Equations

2.1 definition of slope.

In linear equations of the form y = mx + b, where "m" represents the slope, it determines the rate at which the dependent variable (y) changes concerning the independent variable (x). A positive slope indicates an upward incline, while a negative slope represents a downward incline. A slope of zero corresponds to a horizontal line.

2.2 Calculating Slope

To calculate the slope between two points (x₁, y₁) and (x₂, y₂), we use the formula: m = (y₂ - y₁) / (x₂ - x₁). This formula allows us to find the change in y divided by the change in x.

2.3 Interpretation of Slope

The slope's value provides crucial insights into the relationship between variables. A steep slope implies a rapid change, indicating a strong correlation, while a gentle slope signifies a slower change and a weaker correlation. A zero slope denotes a constant relationship, regardless of the independent variable's variations.

Homework 1: Exploring Linear Equations and Slope

In Homework 1, we will dive into various linear equations, both in standard and slope-intercept form, and examine their slopes to gain a better understanding of their properties.

3.1 Solving for Slope in Equations

To solve for the slope in a given linear equation, we first need to identify the value of "m" in the equation y = mx + b. Once we have found the slope, we can interpret its significance and the relationship between the variables.

3.2 Graphing Linear Equations

Graphing linear equations helps visualize their slopes and understand how they translate into lines on the coordinate plane. By plotting the points and connecting them, we gain a visual representation of the equation and its slope.

Answer Key for Homework 1

Here is the step-by-step solution and graphical representation for each linear equation in Homework 1:

4.1 Step-by-Step Solutions

Equation: y = 2x + 3

• Slope (m) = 2
• Step-by-step solution: [Explanation of solving the equation]

Equation: y = -3x + 5

• Slope (m) = -3

4.2 Graphical Representations

• Graph: [Description of the graph]

Practical Applications of Linear Equations and Slope

Linear equations and slope have widespread applications in various fields:

5.1 Real-life Examples

Let's consider a scenario where a small business owner, Amy, runs a bakery. Amy sells two types of cakes: chocolate cakes and vanilla cakes. She wants to analyze her sales data to understand the relationship between the number of cakes sold and the total revenue generated.

Amy keeps track of her sales data for a month and records the following information:

• On the first day, she sells 10 chocolate cakes and 15 vanilla cakes, generating $200 in revenue.
• On the second day, she sells 12 chocolate cakes and 18 vanilla cakes, generating $230 in revenue.
• On the third day, she sells 8 chocolate cakes and 14 vanilla cakes, generating $190 in revenue.

To analyze the relationship between the number of cakes sold and the revenue generated, Amy can use linear equations. Let's define the variables:

Let x be the number of chocolate cakes sold. Let y be the number of vanilla cakes sold.

The revenue generated on a particular day (in dollars) can be represented by the equation:

Revenue = 2x + 3y

Now, we can plug in the values from the sales data to create a system of linear equations:

For the first day: Revenue = 2(10) + 3(15) = 20 + 45 = $65

For the second day: Revenue = 2(12) + 3(18) = 24 + 54 = $78

For the third day: Revenue = 2(8) + 3(14) = 16 + 42 = $58

Now, Amy has three data points: (10, 15, 65), (12, 18, 78), and (8, 14, 58). She can use these data points to create a system of linear equations and find the equation of the line that represents the relationship between the number of cakes sold and the revenue generated.

Once she has the equation, she can use it to predict the revenue for different cake sale combinations in the future. This can help her make informed decisions about her bakery business, such as pricing strategies, inventory management, and overall profitability.

5.2 Importance in Various Fields

Linear equations and slope are fundamental concepts in algebra and mathematics that play a crucial role in various fields, including science, engineering, economics, and more. Understanding these concepts is essential for problem-solving and modeling real-world situations. Let's explore their significance:

Modeling Relationships : Linear equations are used to represent relationships between two variables. For instance, in the form "y = mx + b," where "y" and "x" are variables, "m" is the slope, and "b" is the y-intercept, the equation represents a straight line. The slope (m) indicates the rate of change of "y" concerning "x." By analyzing data and fitting a line through it, we can model and predict relationships between different quantities.

Graphical Representation : Graphing linear equations helps in visualizing data and patterns. The slope of the line determines its steepness or inclination. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. A slope of zero represents a horizontal line. The y-intercept represents the value of "y" when "x" is zero, giving an initial point of reference on the graph.

Solving Problems : Linear equations are used to solve various real-life problems. Whether it's calculating cost functions, determining growth rates, or analyzing data trends, linear equations provide a straightforward approach to finding solutions.

Rate of Change and Proportions : The slope of a linear equation represents the rate of change. For example, if the equation represents the relationship between distance and time for a moving object, the slope would be the object's speed or velocity. Furthermore, when dealing with proportions, the slope represents the constant ratio between two variables.

Interpolation and Extrapolation : Linear equations allow us to interpolate, which means estimating values between known data points. Additionally, they enable extrapolation, which means extending the line beyond the given data points to make predictions for values outside the known range.

Optimization : Linear programming is a technique used in optimization problems to find the best outcome in a mathematical model. It involves maximizing or minimizing a linear objective function, subject to linear inequality or equality constraints. Linear programming is widely used in operations research, economics, and engineering.

Physics and Engineering : Many physical phenomena and engineering systems can be approximated using linear relationships. For example, Hooke's law, which describes the relationship between the force applied to a spring and its resulting displacement, is a linear equation.

Economics : In economics, linear demand and supply functions are often used to model the relationship between price and quantity. The slope of these functions has economic interpretations, such as price elasticity of demand and supply.

In summary, linear equations and slope are essential tools for understanding, analyzing, and predicting relationships between variables in various disciplines. They provide a simple yet powerful framework for problem-solving and decision-making in real-world scenarios.

Common Mistakes and Troubleshooting

In learning about linear equations and slope, some common mistakes can occur. Understanding these errors and how to troubleshoot them will improve the understanding of the subject.

In conclusion, linear equations and slope are foundational concepts in algebra that allow us to analyze the relationships between variables. By understanding slope and its significance, we can interpret various real-life scenarios, making this knowledge highly valuable in multiple fields.

FAQs After The Conclusion

• What is the significance of the slope in a linear equation?
• How do you calculate the slope between two points?
• Can a linear equation have a slope of zero?
• What are some real-life applications of linear equations?
• How can understanding slope help in graphing linear equations?

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Point Slope Form

6 min read • december 10, 2021

Haseung Jun

Standard, Point-Slope and Slope-Intercept Line Equations

Welcome to Algebra, a world where letters and numbers intermingle together to... maybe make your head spin in class once in a while. Crazy, right?! One of the most fundamental concepts in Algebra is the equation of a line. Technically speaking, it has three forms: the point-slope form , the slope-intercept form (y = mx + b), and the standard form (Ax + By = C). In this guide, we'll be focusing on the first (point-slope) form; you'll be able to approach the point-slope form and appreciate how easy it actually is! 📈

Image Courtesy of Pinterest

The Equation

The Point-Slope Form looks like this:

y-y1=m(x-x1)

Many people prefer the point-slope form because it’s super easy to use and the potential of making an error is very minimal, which is good (aka less chances to mess up on your quiz or test)! ❌

This form is also super advantageous because you only need the slope and a point of the line or just two points in the Cartesian plane in order to form the equation! How fun is THAT? 😆

With this form, you can convert to slope-intercept form, standard form, or pretty much any linear equation form of your choice with a little bit of algebra! Doesn’t that sound so fun? Let’s see how we can use it! 🎠

How to Use the Point-Slope Form

Using the equation isn’t that hard, though at first you might scratch your head and wonder, “why are there two x and y s ?” 🤔 Don’t get intimidated just because there are two variables ( x and y ). Looking back to the formula above, you'll plug in numbers for y1 and x1 and end up with the form you're used to. Just remember: don’t mess with x and y ! Only mess with the ones that have a 1 attached to it. Let's repeat that one more time: don't mess with x and y ; only mess with x1and  y1! 🧩

When you’re doing these problems, you’ll usually be given the slope of the line and a point. 

(1) What you’ll do with that information is you’ll first plug in the x -coordinate for x1and the y -coordinate for y1 . Remember that since the equation already contains a subtraction, if your x -coordinate is positive, you should leave it as a subtraction expression, for example, x-3 . If your x -coordinate is negative, it should be x+3 . 🔌

(2) Then, you’ll plug in the slope for m because "m" in the equation represents the slope of the line (why? well, m supposedly stands for either modulus or monter, which is a French word, but no one knows for sure. ⛰️

(3) Anyway, aside from the “where did m come from” question, the next step would be to… do nothing. That’s right! Do NOTHING! 😳 Strange as it sounds, you can leave it as it is. For example, you can leave it as y-4=2(x-3) and you’ll be totally fine. Of course, if your teacher tells you to simplify it into a different form, you’ll have to work it out. 

But what does this mean? Well, let’s first look at an example and work our way through. 🙌

Find the equation of a line that has the slope of 2 and passes through (3, 4).

So, how would you solve this problem? First of all, let’s look at the equation. 🔎

(1) You have the right equation. So far, so good! Then, let’s look at the x- and y-coordinates . It’s 3 and 4, respectively. You then plug in 3 for x1 and 4 for y1 . You should have an equation that looks something like this afterwards:

(2) Then look at the slope . What is it? It’s 2. So plug in 2 for m and you’ll have the equation look like this:

Woohoo! We just found the equation! How fun and cool! 😎 If you're allowed (and if the directions say so), you can leave it as it is in this form. But unfortunately, most teachers will ask you to either change it to slope-intercept form or standard form. 

Here’s the process to simplify into slope-intercept form (y = mx + b). Again, here's the equation in its good ol' point-slope form:

(1) Distribute the 2 to x and -3. 

(2) Then, add 4 to both sides . 

And tada! Only a few steps in and you already have the equation converted into slope-intercept form! 🎉

Now if we’re going to convert it into standard form, it’ll take us a little longer. 

(1) Subtract 2x from both sides . The 2x on the right side will be cancelled out.

(2) Then, rearrange the order of x and y . 🚲

(3) Almost there! Since we can’t let A from Ax + By = C (standard form equation) be negative, we’ll have to multiply the equation by -1 . 💥

So voila ! Final result for standard form! 😍

Wait, What if Slope is Not Given?

Sometimes, teachers won't give you the slope of the line. Don’t worry, it's not the end of the world yet! Instead of the slope, they’ll provide you with two points. It might end up taking a bit more work on your end, but oh well, school is school. 🏫

In this case, you’ll use the slope formula [insert link here] to find the slope. Remember what the slope formula is? The " rise-over-run" formula?

Now with this equation, you’ll just use the two x- and y- coordinates to find the slope ! When you’re plugging in your coordinates into the equation, you get the freedom to choose whichever point you want to use! 🌟

Let’s look at an example!

Find the equation of a line that passes through (6, 1) and (4, 5).

So in this case, you don’t have a slope. Again, don’t panic! Just plug it into the slope formula and you’ll have it! 😉

(1) With the slope formula, plug in the coordinates for the appropriate variable!

(2) When you subtract , you should get this:

Since 4 is divisible by -2,

Your slope (m) should equal -2 ! 

(3) Not hard right? Then all you have to do is plug it in for the equation . 

y-1= -2 (x-6)

And that’s it! Hurray! 

If you convert it to slope-intercept form , it should look like:

y-4 = -2x + 12

y = -2x + 16

If you convert it to standard form , it should look like:

y= -2x + 16

2x + y = 16

That’s it for this example. You're done! 😝

Wrapping Up: Point-Slope Who?!

As you can see, using, finding, and applying point-slope form is actually super fun and straightforward with practice! It’s really easy to use; in fact, most people use the point-slope form of an equation to change it to its other forms. ➡️

Feeling hyped up now? Go finish your math homework! ✨

(Image courtesy of Quote Master )

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Chapter 3: Graphing

3.4 Graphing Linear Equations

There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation.

If the equation is given in the form [latex]y = mx + b[/latex], then [latex]m[/latex] gives the rise over run value and the value [latex]b[/latex] gives the point where the line crosses the [latex]y[/latex]-axis, also known as the [latex]y[/latex]-intercept.

Example 3.4.1

Given the following equations, identify the slope and the [latex]y[/latex]-intercept.

• [latex]\begin{array}{lll} y = 2x - 3\hspace{0.14in} & \text{Slope }(m)=2\hspace{0.1in}&y\text{-intercept } (b)=-3 \end{array}[/latex]
• [latex]\begin{array}{lll} y = \dfrac{1}{2}x - 1\hspace{0.08in} & \text{Slope }(m)=\dfrac{1}{2}\hspace{0.1in}&y\text{-intercept } (b)=-1 \end{array}[/latex]
• [latex]\begin{array}{lll} y = -3x + 4 & \text{Slope }(m)=-3 &y\text{-intercept } (b)=4 \end{array}[/latex]
• [latex]\begin{array}{lll} y = \dfrac{2}{3}x\hspace{0.34in} & \text{Slope }(m)=\dfrac{2}{3}\hspace{0.1in} &y\text{-intercept } (b)=0 \end{array}[/latex]

When graphing a linear equation using the slope-intercept method, start by using the value given for the [latex]y[/latex]-intercept. After this point is marked, then identify other points using the slope.

This is shown in the following example.

Example 3.4.2

Graph the equation [latex]y = 2x - 3[/latex].

First, place a dot on the [latex]y[/latex]-intercept, [latex]y = -3[/latex], which is placed on the coordinate [latex](0, -3).[/latex]

Now, place the next dot using the slope of 2.

A slope of 2 means that the line rises 2 for every 1 across.

Simply, [latex]m = 2[/latex] is the same as [latex]m = \dfrac{2}{1}[/latex], where [latex]\Delta y = 2[/latex] and [latex]\Delta x = 1[/latex].

Placing these points on the graph becomes a simple counting exercise, which is done as follows:

Once several dots have been drawn, draw a line through them, like so:

Note that dots can also be drawn in the reverse of what has been drawn here.

Slope is 2 when rise over run is [latex]\dfrac{2}{1}[/latex] or [latex]\dfrac{-2}{-1}[/latex], which would be drawn as follows:

Example 3.4.3

Graph the equation [latex]y = \dfrac{2}{3}x[/latex].

First, place a dot on the [latex]y[/latex]-intercept, [latex](0, 0)[/latex].

Now, place the dots according to the slope, [latex]\dfrac{2}{3}[/latex].

This will generate the following set of dots on the graph. All that remains is to draw a line through the dots.

The second method of drawing lines represented by linear equations and functions is to identify the two intercepts of the linear equation. Specifically, find [latex]x[/latex] when [latex]y = 0[/latex] and find [latex]y[/latex] when [latex]x = 0[/latex].

Example 3.4.4

Graph the equation [latex]2x + y = 6[/latex].

To find the first coordinate, choose [latex]x = 0[/latex].

This yields:

[latex]\begin{array}{lllll} 2(0)&+&y&=&6 \\ &&y&=&6 \end{array}[/latex]

Coordinate is [latex](0, 6)[/latex].

Now choose [latex]y = 0[/latex].

[latex]\begin{array}{llrll} 2x&+&0&=&6 \\ &&2x&=&6 \\ &&x&=&\frac{6}{2} \text{ or } 3 \end{array}[/latex]

Coordinate is [latex](3, 0)[/latex].

Draw these coordinates on the graph and draw a line through them.

Example 3.4.5

Graph the equation [latex]x + 2y = 4[/latex].

[latex]\begin{array}{llrll} (0)&+&2y&=&4 \\ &&y&=&\frac{4}{2} \text{ or } 2 \end{array}[/latex]

Coordinate is [latex](0, 2)[/latex].

[latex]\begin{array}{llrll} x&+&2(0)&=&4 \\ &&x&=&4 \end{array}[/latex]

Coordinate is [latex](4, 0)[/latex].

Example 3.4.6

Graph the equation [latex]2x + y = 0[/latex].

[latex]\begin{array}{llrll} 2(0)&+&y&=&0 \\ &&y&=&0 \end{array}[/latex]

Coordinate is [latex](0, 0)[/latex].

Since the intercept is [latex](0, 0)[/latex], finding the other intercept yields the same coordinate. In this case, choose any value of convenience.

Choose [latex]x = 2[/latex].

[latex]\begin{array}{rlrlr} 2(2)&+&y&=&0 \\ 4&+&y&=&0 \\ -4&&&&-4 \\ \hline &&y&=&-4 \end{array}[/latex]

Coordinate is [latex](2, -4)[/latex].

For questions 1 to 10, sketch each linear equation using the slope-intercept method.

• [latex]y = -\dfrac{1}{4}x - 3[/latex]
• [latex]y = \dfrac{3}{2}x - 1[/latex]
• [latex]y = -\dfrac{5}{4}x - 4[/latex]
• [latex]y = -\dfrac{3}{5}x + 1[/latex]
• [latex]y = -\dfrac{4}{3}x + 2[/latex]
• [latex]y = \dfrac{5}{3}x + 4[/latex]
• [latex]y = \dfrac{3}{2}x - 5[/latex]
• [latex]y = -\dfrac{2}{3}x - 2[/latex]
• [latex]y = -\dfrac{4}{5}x - 3[/latex]
• [latex]y = \dfrac{1}{2}x[/latex]

For questions 11 to 20, sketch each linear equation using the [latex]x\text{-}[/latex] and [latex]y[/latex]-intercepts.

• [latex]x + 4y = -4[/latex]
• [latex]2x - y = 2[/latex]
• [latex]2x + y = 4[/latex]
• [latex]3x + 4y = 12[/latex]
• [latex]4x + 3y = -12[/latex]
• [latex]x + y = -5[/latex]
• [latex]3x + 2y = 6[/latex]
• [latex]x - y = -2[/latex]
• [latex]4x - y = -4[/latex]

For questions 21 to 28, sketch each linear equation using any method.

• [latex]y = -\dfrac{1}{2}x + 3[/latex]
• [latex]y = 2x - 1[/latex]
• [latex]y = -\dfrac{5}{4}x[/latex]
• [latex]y = -3x + 2[/latex]
• [latex]y = -\dfrac{3}{2}x + 1[/latex]
• [latex]y = \dfrac{1}{3}x - 3[/latex]
• [latex]y = \dfrac{3}{2}x + 2[/latex]
• [latex]y = 2x - 2[/latex]

For questions 29 to 40, reduce and sketch each linear equation using any method.

• [latex]y + 3 = -\dfrac{4}{5}x + 3[/latex]
• [latex]y - 4 = \dfrac{1}{2}x[/latex]
• [latex]x + 5y = -3 + 2y[/latex]
• [latex]3x - y = 4 + x - 2y[/latex]
• [latex]4x + 3y = 5 (x + y)[/latex]
• [latex]3x + 4y = 12 - 2y[/latex]
• [latex]2x - y = 2 - y \text{ (tricky)}[/latex]
• [latex]7x + 3y = 2(2x + 2y) + 6[/latex]
• [latex]x + y = -2x + 3[/latex]
• [latex]3x + 4y = 3y + 6[/latex]
• [latex]2(x + y) = -3(x + y) + 5[/latex]
• [latex]9x - y = 4x + 5[/latex]

Answer Key 3.4

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Linear Equations

A linear equation is an equation for a straight line

These are all linear equations:

Let us look more closely at one example:

Example: y = 2x + 1 is a linear equation:

The graph of y = 2x+1 is a straight line

• When x increases, y increases twice as fast , so we need 2x
• When x is 0, y is already 1. So +1 is also needed
• And so: y = 2x + 1

Here are some example values:

Check for yourself that those points are part of the line above!

Different Forms

There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").

Examples: These are linear equations:

But the variables (like "x" or "y") in Linear Equations do NOT have:

• Exponents (like the 2 in x 2 )
• Square roots , cube roots , etc

Examples: These are NOT linear equations:

Slope-intercept form.

The most common form is the slope-intercept equation of a straight line :

Example: y = 2x + 1

• Slope: m = 2
• Intercept: b = 1

Point-Slope Form

Another common one is the Point-Slope Form of the equation of a straight line:

Example: y − 3 = (¼)(x − 2)

It is in the form y − y 1 = m(x − x 1 ) where:

General Form

And there is also the General Form of the equation of a straight line:

Example: 3x + 2y − 4 = 0

It is in the form Ax + By + C = 0 where:

There are other, less common forms as well.

As a Function

Sometimes a linear equation is written as a function , with f(x) instead of y :

And functions are not always written using f(x):

The Identity Function

There is a special linear function called the "Identity Function":

And here is its graph:

It is called "Identity" because what comes out is identical to what goes in:

Constant Functions

Another special type of linear function is the Constant Function ... it is a horizontal line:

No matter what value of "x", f(x) is always equal to some constant value.

Using Linear Equations

You may like to read some of the things you can do with lines:

• Finding the Midpoint of a Line Segment
• Finding Parallel and Perpendicular Lines
• Finding the Equation of a Line from 2 Points

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Unit 3: Linear relationships

Lesson 3: representing proportional relationships.

• Graphing proportional relationships: unit rate (Opens a modal)
• Graphing proportional relationships from a table (Opens a modal)
• Graphing proportional relationships from an equation (Opens a modal)
• Graphing proportional relationships Get 3 of 4 questions to level up!

Lesson 4: Comparing proportional relationships

• Rates & proportional relationships example (Opens a modal)
• Rates & proportional relationships: gas mileage (Opens a modal)
• Rates & proportional relationships Get 5 of 7 questions to level up!

Lesson 7: Representations of linear relationships

• Linear & nonlinear functions: missing value (Opens a modal)

Lesson 8: Translating to y=mx+b

• Intro to slope-intercept form (Opens a modal)
• Graph from slope-intercept equation (Opens a modal)

Lesson 9: Slopes don't have to be positive

• Intro to intercepts (Opens a modal)
• Slope-intercept equation from slope & point (Opens a modal)
• Linear & nonlinear functions: word problem (Opens a modal)
• Intercepts from a graph Get 3 of 4 questions to level up!
• Slope from graph Get 3 of 4 questions to level up!
• Slope-intercept intro Get 3 of 4 questions to level up!
• Graph from slope-intercept form Get 3 of 4 questions to level up!
• Slope-intercept equation from graph Get 3 of 4 questions to level up!

Lesson 10: Calculating slope

• No videos or articles available in this lesson
• Slope from two points Get 3 of 4 questions to level up!

Lesson 11: Equations of all kinds of lines

• Converting to slope-intercept form (Opens a modal)

Extra practice: Slope

• Intro to slope (Opens a modal)
• Worked examples: slope-intercept intro (Opens a modal)
• Graphing slope-intercept form (Opens a modal)
• Writing slope-intercept equations (Opens a modal)
• Slope-intercept form review (Opens a modal)
• Slope-intercept from two points Get 3 of 4 questions to level up!

Lesson 12: Solutions to linear equations

• Solutions to 2-variable equations (Opens a modal)
• Worked example: solutions to 2-variable equations (Opens a modal)
• Solutions to 2-variable equations Get 3 of 4 questions to level up!

Lesson 13: More solutions to linear equations

• Completing solutions to 2-variable equations (Opens a modal)
• Complete solutions to 2-variable equations Get 3 of 4 questions to level up!

Extra practice: Intercepts

• x-intercept of a line (Opens a modal)
• Intercepts from an equation (Opens a modal)
• Worked example: intercepts from an equation (Opens a modal)
• Intercepts of lines review (x-intercepts and y-intercepts) (Opens a modal)
• Intercepts from an equation Get 3 of 4 questions to level up!