The topic of linear equations is perhaps the single most important one on the SAT math section. Line equations and their graphs are all over this test. As a high-school student, linear equations were expected to have been the rudiment of your math education, they're applicable to a number of real-life situations, and there's a plethora of things you can ask about the topic. I'd advise you to know linear equations very, very well before taking the SAT.

Although there are many different kinds of line questions on the SAT, there's one which is almost certain to appear at least once, and maybe more, on your test. Even without understanding too much about lines, after reading this article you should be able to save yourself some time and frustration, and answer this question type in a couple of seconds.

This question asks if you understand the parameters of the line equation. Meaning, given the parameters of a line - given the y-intercept, say, or the slope - can you explain what they represent in the real life situation?

The answer is, in short, yes. Line parameters always mean the same thing, so you shouldn't have to think anew each time you're presented with the question.

There are two parameters in each line equation, y-intercept and slope. Today we'll discuss the significance of the y-intercept, and how that appears in questions on the SAT. In a future article, we'll discuss this question type in regard to slope.

Let's start by briefly discussing the fundamentals of the line equation itself. We'll then move on to the specific question type we're meant to discuss today.

The line equation y = m*x + b

As you hopefully already know, the above equation is what is commonly termed the line equation. 'b' in this case represents the y-intercept, and 'm' represents slope. "Finding the equation" of a specific line really means finding its y-intercept and its slope. Once I know those two things, I've "found" the line equation.

Of course, the letters can change; the y-intercept, for instance, can be represented by a, s, or some other random letter. It's just a symbol.

Let's look at a general line:

b, the y-intercept, is where the line crosses the y axis. In this case, b is 5.

We'll discuss the meaning of slope in a later article.

The physical meaning of line parameters

Note that b is the value of the graph when x = 0. That is, b is the value we're starting at.

The value 'we're starting at' can translate into several things in real-life situations. Here are a few examples:

head start

down payment

starts at, opens with, commences with

flat fee, initial cost, one-time fee

etc.

Let's explain why terms like 'flat fee' or 'down payment' are included in this list. Take, for example, a consulting firm which charges a flat fee of $500 simply to hire them. They then charge an hourly fee of $150 for every hour of consultation you receive.

Before you've even clocked hours, your fee is $500. That is, the fee when x = 0 hours is $500. That's the significance of the y-intercept.

In a moment we'll look at some examples. In many of them, you may note that there's plenty of extra detail unnecessary to answer the question. That's because these are easy questions masquerading as difficult ones. The goal is to confuse you by dangling irrelevant details, so that you don't know where to look.

You shouldn't have to read any of those extra details. In fact, some of these questions cite confusing studies, or introduce concepts from physics and chemistry. You don't have to understand any of it in order to answer the question correctly. And you shouldn't try. As soon as you see the graph and/or the equation of a line, skip right to the question. Are they asking what starting value some function had? For a one-time fee, or for the head start? Then the answer's the b-intercept.

Alternatively, they may give you the equation of a line, and ask what the y-intercept represents. In that case, simply search the answers for an appropriate keyword.

Examples

Example #1

Given: y = 55 + 15*x

The equation above represents the total cost, y, that a moving company charges customers for its services. x represents the number of hours the moving company worked for the customer. The total cost y represent a flat fee, as well as an hourly rate for each hour of work. The units of x are hours, and the units of y are dollars.

Question #1

When the equation for total cost is graphed, what is effectively represented by the y-intercept of the equation?

A) A flat fee of 55

B) Total daily charges of $70

C) A charge per mile of $15

D) A charge per hour of $55

The instant you see a linear equation and a lengthy verbal description, skip the description and go to the question itself. The question is asking about y-intercept, so glance through the answers for the right keyword. It's in answer A) - "a flat fee."

Note that you must be able to identify linear equations just by glancing at them. If you can't do that yet, practice on some worksheets until you have it.

Example #2

grid question (no answer options):

Shirley opened a small online business. She started the business with 25 products and drafted a plan to add 4 new products per month. Due to some financial difficulties, she decided instead to add 5 new products every three months. The function y = a*x + b represents how many products Shirley was selling, x months after she first opened her online company.

Question #2

What is the value of b?

This is a linear equation, since the function that represents the number of Shirley's products is

y = a*x +b. The question is asking for the value of 'b', which here plays the part of the y-intercept. In other words, they're asking for the number of products Shirley started with. Now look at the second sentence: "She started the business with 25 products." The answer is 25.

Example #3

Elliott handles complaints for an insurance claims company. He receives a fresh batch of complaints each day and is able to process a certain amount per hour. In consequence, the number of complaints Elliott has to work through after the passage of x hours can be modeled by the equation f(x) = 17 - 2*x, where x is the number of hours, and y is the total number of complaints.

Question #3

What does the value of 17 signify in the equation?

A) Elliott processes 17 complaints per day.

B) Elliott processes 17 complaints per hour.

C) Elliott opens the day with 17 complaints.

D) Elliott is able to process 1 complaint in 17 days.

Since 17 is the y-intercept in this question, we'll choose the answer with the appropriate keyword. This is answer C): 'Elliott opens the day with 17 complaints.' Remember, the y-intercept is what you're starting with.

Note that both A) and B) use the word 'per': 'per day', 'per hour'. This is language reserved for slope, not for the y-intercept. We'll discuss that more in our article on real-life questions involving slope.

Example #4

The following graph shows the total cost T, in dollars, of ice skating in the Willoughby ski rink for x hours:

Question #4

What does the T-intercept represent in the graph?

A) The total number of hours skied.

B) The total amount of people skiing.

C) The cost of skiing per hour

D) The entrance fee for the ski rink.

The 'T-intercept' is another word for the y-intercept, since this function is represented by 'T' rather than y. The appropriate key word appears in D): 'The intial cost of entering the ski rink.' That is, when '0' hours have passed, the client will already have paid the price of the T-intercept: $5.00

Example #5

Klonder Amusement Park charges a one time service charge for using their facilities, as well as an additional fee for any attraction visited. The total amount, in dollars, any visitor will pay to visit n attractions is represented by the equation y = 4*n + 15.

Question #5

What does the value 15 signify?

A) The total amount, in dollars, a visitor will pay to visit n attractions.

B) The price a visitor will pay for a single attraction, in dollars.

C) The amount of the service charge, in dollars.

D) The total amount, in dollars, a visitor will pay over time.

Looking at the equation, we see that 15 is the y-intercept. So we're looking for the amount a visitor will pay when he has visited n=0 attractions. Words like "total amount" or "price for a single" have nothing to do with the y-intercept. The only option which works is C), the amount of the service charge. And if you're wondering what the service charge is, glance back at the question. The first sentence mentions a 'one time service charge.' This is exactly what we're looking for. The answer is C).

Example #5 Elise is studying the velocity of a moving object in meters per second, where acceleration is constant. The velocity of the object she's studying, as a function of time, can be represented by the following equation: v = 35.4 + 5.4*t, where t is in units of seconds.

Question #5

What does 35.4 represent in this equation?

A) The object's velocity at 5.4 seconds, in meters per second.

B) The increase in the object's velocity, in meters per second, corresponding to an increase of 5.4 seconds.

C) The object's velocity at 0 seconds.

D) The increase in the object's velocity, in meters per second, corresponding to an increase of 35.4 seconds.

35.4 is the y-intercept of the equation. That is, it's the value of the function (velocity) when the x axis (time) is 0. This is represented by option C).

Don't be intimidated by the formula. Although this is a question taken from the realm of physics, you need zero knowledge of science to be able to answer it. Ignore the background, skip to the equation itself, skim the answer options for the right words, and move on.

Note that all the options, like in questions before, list units: 'meters per second'. The extra words make the answer options seem more convoluted than they really are. There are question types on the SAT where wrong units can lead to a wrong answer, but this isn't one of them. You can ignore them.

In addition, note the use of the word 'increase': 'increase in the speed of sound' in options B) and D). This is a keyword which denotes slope, not the y-intercept. We'll discuss this more in our article on the significance of slope.

Example #6

Rosemarie regularly walks her sister Alice to school in the mornings. This morning, she's running late, so she allows her sister to get a head start. She eventually catches up to Alice as Alice is passing the bakery. It takes Alice a total of 10 minutes to reach the bakery, and it takes Rosemarie 4 minutes.

The graph below show Alice and Rosemarie's progress, where x measures minutes and the y axis measures how much Alice and Rosemarie have walked. The graph modeling Alice's progress is in blue, and the graph modeling Rosemarie's progress is in green.

Question #6 What value shows Alice's head start?

A) 10

B) 12

C) 6

D) 15

"Head start" refers to where Alice was when Rosemarie started walking - that is, when x=0. In other words, they're asking for the y-intercept of Alice's graph. The answer is B), 12.