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SAT
Course: SAT > Unit 6
Lesson 3: Heart of Algebra: lessons by skill- Solving linear equations and linear inequalities | Lesson
- Understanding linear relationships | Lesson
- Linear inequality word problems | Lesson
- Graphing linear equations | Lesson
- Systems of linear inequalities word problems | Lesson
- Solving systems of linear equations | Lesson
- Systems of linear equations word problems | Lesson
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Understanding linear relationships | Lesson
This lesson covers the following OSP skills:
Linear equation word problems
Linear function word problems
Interpreting linear functions
What are linear relationships, and how frequently do they appear on the test?
A linear relationship is any relationship between two variables that creates a line when graphed in the x, y-plane. Linear relationships are very common in everyday life.
Linear relationships appear frequently on the SAT: about 25, percent of the SAT Math test involves linear relationships! For this reason, we recommend you get comfortable working with linear relationships in their many forms.
In this lesson, we'll:
- Review the basics of linear relationships
- Practice writing linear equations based on word problems
- Identify the important features of linear functions
On your official SAT, you'll likely see 5 to 7 questions like those found in this lesson. The skills covered here will also be important for the following lessons:
You can learn anything. Let's do this!
Linear relationships
Linear equations can be used to represent the relationship between two variables, most commonly x and y. To form the simplest linear relationship, we can make our two variables equal:
By plugging numbers into the equation, we can find some relative values of x and y.
| x | y |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
If we plot those points in the x, y-plane, we create a line.
Every possible linear relationship is just a modification of this simple equation. We might multiply one of the variables by a coefficient or add a constant to one side of the equation, but we'll still be creating a linear relationship.
How do we translate word problems into linear equations?
Modeling real world scenarios
Translating word problems
It may not be hard to translate "Maya is 3 inches taller than Geoff" into a linear equation, but some SAT word problems are several sentences long, and the information we need to build an equation may be scattered around.
Let's look at some examples!
A car with a price of dollar sign, 17, comma, 000 is to be purchased with an initial payment of dollar sign, 5, comma, 000 and monthly payments of dollar sign, 240. Which of the following equations can be used to find the number of monthly payments, m, required to complete the purchase, assuming there are no taxes or fees?
The width of a rectangular vegetable garden is w feet. The length of the garden is 8 feet longer than its width. Which of the following expresses the perimeter, in feet, of the vegetable garden in terms of w ?
The concession stand at a high school baseball game sold bags of peanuts for dollar sign, 2, point, 50 each and hot dogs for dollar sign, 3, point, 00 each. If the concession stand brought in dollar sign, 196 and sold 42 hot dogs, how many bags of peanuts did the concession stand sell?
What will we be asked to do in linear equations word problems?
On the test, we may be asked to:
- Write our own equation based on the word problem (frequent)
- Write our own equation and then solve it (frequent)
- Solve a given equation based on the word problem (infrequent)
Try it!
What are important features of linear functions?
Linear equations in slope-intercept form
Linear functions
Any linear equation with two variables is technically a function. Linear functions are usually written in either slope-intercept form or standard form. We need a thorough and flexible understanding of these forms in order to approach many SAT questions about linear relationships.
Slope-intercept form
The slope-intercept form of a linear function, start color #ca337c, y, equals, m, x, plus, b, end color #ca337c, where m and b are constants, tells us both the slope and the y-intercept of the line:
- The slope is equal to m.
- The y-intercept is equal to b.
Standard form
The standard form of a linear function, start color #ca337c, A, y, plus, B, x, equals, C, end color #ca337c, where, A, B, and C are constants, will often be used in word problem scenarios that have two inputs, instead of an input and an output. To find the slope or y-intercept of a line in standard form, it's often most convenient to convert the equation to slope-intercept form by isolating y.
What will we be asked to do in linear function word problems?
On the test, we may be asked to:
- Write our own linear function based on the word problem (We may need to calculate the slope or y-intercept in more challenging questions.)
- Identify the meaning of a value in a given function that models a scenario
Try it!
Your turn!
Things to remember
The slope-intercept form of a linear equation, y, equals, m, x, plus, b, tells us both the slope and the y-intercept of the line:
- The slope is equal to m.
- The y-intercept is equal to b.
We can write the equation of a line as long as we know either of the following:
- The slope of the line and a point on the line
- Two points on the line
Want to join the conversation?
I would like to make a request, could you please add a video that explains what exactly is the slope-intercept form?(22 votes)
I'd like to direct you to a Khan Academy video that does in fact do this. You'll find that a lot of the fundamental skills for SAT Math can be learned elsewhere on Khan Academy, such as what slope-intercept from is.
https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:intro-to-slope-intercept-form/v/slope-intercept-form(25 votes)
- Hello
I have just started studying SAT Math Test lessons and it was my first lesson (Understanding liner relationships)
-How can I more practice the mentioened lesson?
This is very important to say I finished my High-School around 10 years ago and there is a huge gap! So If I study and finish all 40 lessons in SAT Math Test lesson part that you have mentioned them will be enough to get ready for SAT Test Math part?(9 votes)
Hello,
Khan Academy also has a bunch of SAT practice (the skill practices should cover the same topics as the videos) and full-length practice tests. If you take a diagnostic math quiz or a full test, Khan Academy will recommend certain practices and help you focus on the areas you need to work on... which is probably easier than studying by going through every lesson. :) I hope this helps. Good luck on the SAT!
https://www.khanacademy.org/mission/sat/practice/math
https://www.khanacademy.org/mission/sat/exams(10 votes)
- I have a confusion in the last question. Why we are choosing 0 as y2 and 10 as y1? Why can`t we reverse them?
What I mean to say is, if we create a graph for the values then it would be a negative graph(with a negative slope).So 330,000 should be x1 and vice versa.
can anyone explain this?(3 votes)- You very well could reverse them, and it should give you the same slope. The slope formula doesn't have to use an earlier point on the line and a later point. Instead, it is just any two points.
Typically, you would have the earlier time as t1 and the later time as t2 because that makes chronological sense, but this is the SAT, so you if you can cut out a step or two in the interest of saving time, that's good.
(330,000 - 80,000) / (0 - 10) is at least to me, visually a lot easier to do than (80,000 - 330,000) / (10 - 0) because negatives with big numbers can be scary if you're in a hurry.
It's a matter of personal preference whether you want to always subtract the latter data point from the former or mix it up to fit the situation, do whichever one feels more natural to you.(5 votes)
- I am confused about the formula taught in algebra 1 for linear relationships with "n" in it.(5 votes)
i need to know the easiest way to solve linear equations i'm really slow with solving them(2 votes)- could you make a video that explains how two equations can have no solution?(1 vote)
You might find these resources on Khan Academy helpful: article "Number of solutions to system of equations review" and video "Systems of equations number of solutions: fruit prices (1 of 2)".
Just put those titles (or anything you want more info on) into the search bar at the top of Khan's webpages, and you can find some useful things!
If you can't find the material you want by searching there, make sure to put any content requests in the Support Community. The link to it's at the bottom of the page!(3 votes)
- The equation above models yyy, the amount in dollars charged by a website hosting service to host a website for mmm months. The total cost consists of a one-time setup fee plus a monthly charge for hosting. When the equation is graphed in the xyxyx, y-plane, what does the yyy-intercept of the graph represent in terms of the model?
Choose 1 answer:
Choos(2 votes) - How are you? I hope you're okay(2 votes)
- but with the lemonade example, mustn't it start at a negative y, cos before they sell any lemonade they are -25 and want to reach a break even (the x axis) after 25?(1 vote)
- The problem doesn't ask for a graph, only an answer for the number of cups it would take to break even. This means that you can draw the graph however you want to. The way it's done in the video is with the y axis being the total amount of money coming in from the glasses of lemonade. This means that to break even, they would have to make $25 worth of glasses, or find where the line hits y = 25.
For the graph you suggest, it's also correct. However, in this case, the y axis would measure the profit of the lemonade stand instead of the gross revenue. In this case, the line would start at -25 and then the break even point would be y = 0 like you say. Both approaches are equally correct.(2 votes)
- Hi,
I understand that linear relationships must have 1 or two variables in the numerator, degree 1 and when graphed a straight line. However, when we have S = D/t the t is in the denominator but it is still a linear relationship. How is it possible in this case that a variable is in the denominator?(1 vote)- Well, you could for instance just multiply the t to both sides of the equation and boom, now you have S * t = D, which you can see is a linear relationship where D is your output and depending on how the question is phrased, S and t are either your slope and your input or your input and your slope, respectively.
You're right that when we have a variable in the denominator, the relationship is nonlinear. However, in some problems, t might not be a variable, and instead could be a part of the slope term. Let's say you were running, and measuring the distance, speed, and duration of your run. If you were specifically looking at Speed versus Distance for a constant amount of time each time you run, you would get a linear result. If you were looking at the Speed when varying the duration of your run for a set amount of distance, you would get a nonlinear result because now the thing you're varying is in the denominator. Does that clear it up somewhat?(2 votes)