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Linear Expressions, Equations, and Functions

SAT Math Overview. Topic: Linear Expressions, Equations, and Functions.

Overview

Linear Expressions, Equations, and Functions are found throughout the SAT, so it’s very important to be comfortable with all the key aspects of linear functions and equations. In particular, make sure you know these concepts:

  • Linear Expressions
  • Linear Equations
  • Linear Functions 
    • Slope-Intercept Form
    • Point-Slope Form
  • Slope
  • Parallel/Perpendicular Slope
  • Evaluating Linear Functions
  • Linear Models

 

Let’s look at each of these in more detail:

Linear Expressions

Linear expressions include a variable whose exponent is 1. Here are some examples of linear expressions:

Examples of linear expresssions: negative 2x, 3y, 24z+2, 3x+4y.

Be sure you know how to translate a verbal expression into a linear expression like the ones shown above. To see examples of how to do this, click on this link to see a slide show.

Expressions aren’t equations but they are an important component of linear equations. Also, you can add and subtract linear expressions and still have a linear expression. Multiplying or dividing linear expressions will result in a non-linear expression. Here are some examples:

Adding and subtracting linear expressions results in linear expressions. Multiplying and dividing linear expressions results in non-linear expressions.

 

SAT Skill: Interpreting Expressions

Not only do you need to be comfortable manipulating linear expressions, you need to be able to interpret what the terms of an expression mean. Here are some examples:

Example 1

It costs $24/hour to rent a bike and the total fee is summarized by this expression:

35 plus 24 h

Questions you might be asked:

  • What does h represent? (The number of hours.)
  • What does 24h represent? (The total cost for the hourly rate.)
  • What does 35 represent? (An initial, start-up fee.)

Example 2

A hollow tube of radius r and length l is flattened into a rectangle. What is the perimeter of this rectangle?

 

An illustration of a cylinder in three dimensions and flattened to a two-dimensional figure.

Perimeter is based on the linear expression 2(l + w), where l is the length of the rectangle and w is the width. Find the corresponding expressions for these terms from the diagram. 

The length of the rectangle corresponds to the circumference of the circular part of the cylinder. The width of the rectangle corresponds to the height of the cylinder. The resulting perimeter expression is this:

2 times the expression 2 pi• r + h

Linear Equations

Linear equations include a linear expression equal to a number or another linear expression. A linear equation can have one or more variables, but all terms must be linear.

Here are examples of linear equations with one variable.

Three examples of linear equations.

SAT Skill: Solving Equations

You’ll see a number of problems on the SAT that involve linear expressions and equations. A key skill to have is solving one-variable equations quickly and efficiently. Here are some examples. 

Example 1

 If one third x + one fourth = 1 and k = one half, what is x?

 

This is a one-variable equation with fractions. Start by isolating the x-term on the left side of the equation:

The solution to the equation 1/3 x + one fourth = 1/2.

Then multiply both sides by 3 to isolate x on the left, and simplify to solve.

The solution to the equation 1/3 x + one fourth = 1/2.

Example 2

 If 11 + 3x is 12 more than 20, what is the value of 10x?

Interpret the verbal description to write the equation:

The equation 11 + 3x = 20 + 12

Now solve for x:

The solution to the equation 11+3x=32

Recall that the question asked for 10x:

 

Finding the value of 10x.

 

If you need more practice in solving one-variable equations, click on the following links:

Another skill related to equation solving is the ability to rewrite an equation in an equivalent form. See the examples below.

SAT Skill: Manipulating Equations

In addition to solving linear equations, you need to be able to manipulate these equations to come up with equivalent forms. 

Example 1

Suppose 3x = 27, what is the value of 12x ÷  6?

To get the expression 12x on the left side of the equation, multiply both sides by 4.

Solution to the equation 3 x • 4 equals 27 • 4

Next divide both sides by 6:

Solution to the equation 3 x • 4 equals 27 • 4

Example 2

Suppose 12x = 48, what is the value of 15x + 10?

To get the expression 15x solve for x and multiply by 15:

Solution to the equation 12x = 48.

Now add 10 to both sides of the equation:

Solution to the equation 15x+10=60+10.

 

Make sure you are comfortable with the properties of equality and the structure of an equation. For a quick review, click on this link.

Linear Functions

Before studying what a linear function is, make sure you are comfortable with the following concepts, which we will also review:

  • What a function is
  • Independent variable
  • Dependent variable
  • Domain
  • Range
  • Different representations of functions

Brief Review of Functions 

What Is a Function? A function is a one-to-one mapping of input values (the independent variable) to output values (the dependent variable). Click on this link to see a quick tutorial on what a function is. This slide show goes over the following key points:

  • For every input value (x), there is a unique output value, f(x).
  • Functions can be represented as equations, tables, and graphs.
  • A function machine is a useful visual representation of the input/output nature of functions.

Image of a function machine.

Dependent/Independent Variables. When one variable depends on another, then it is the dependent variable. For example, the faster your speed, the farther you travel. Suppose that speed is represented by the variable s and the distance traveled is represented by the variable d. 

Here’s how to describe the relationship between s and d:

The faster the speed, the more distance traveled.

Distance is dependent on speed.

Distance is a function of speed.

d = f(s)

When studying functions, make sure you are comfortable telling the difference between the independent variable and dependent variable. Get comfortable using function notation. To learn more about function notation, click on this link.

Domain and Range. A function shows the relationship between two variables, the independent variable and the dependent variable. The domain is the allowed values for the independent variable. The range is the allowed values for the dependent variable. The domain and range influence what the graph of the function looks like.

For a detailed review of what domain and range are, click on this link to learn more. You’ll see definitions of the terms domain and range, as well as examples of how to find the domain and range for given functions.

 

Multiple Representations of Functions. We mentioned previously that functions can be represented in different ways. In fact, any function can be represented by an equation, usually f(x) equal to some expression; a table; or a graph. For a detailed review of multiple representations of functions, click on this link, to see a slide show that includes examples of these multiple representations.

Slope-Intercept Form

The most important form of a linear function is the slope-intercept form. 

Slope-Intercept Form. The equation of a linear function written in a form that easily identifies the slope (m) and y-intercept (b).

Given the slope, m, and the y-intercept, b, for a linear function, you can easily construct the equation and graph of the linear function. To see examples of graphing linear functions in slope-intercept form, click on this link. This slide show also includes a video tutorial.

Point-Slope Form

Another method of finding the slope-intercept form involves using the point-slope form. In this case you are given the slope, m, of the line and one set of coordinates, (x, y), on the line. This is what the point-slope form looks like. 


 

A formula used to find the equation of a line in slope-intercept form, for a given set of coordinates and slope.

 

To see examples of deriving the linear function using the  in point-slope form, click on this link. This slide show tutorial walks you through the process and provides several worked-out examples.

A number of SAT questions will test your understanding of linear functions in slope-intercept and point-slope form.

SAT Skill: Deriving a Linear Function

These examples rely on a basic understanding of slope-intercept form.

Example 1

What is the slope-intercept form of the equation of this graph?

 

The graph of a line of slope 2.

You can see from the graph that the y-intercept is 2. 

To determine the slope of the line, follow these steps:

  1. Start at the y-intercept, which is a point on the line.
  2. Count the number of increments up the y-axis and to the right on the x-axis until you are back on the line.
  3.  
  4.  

For this graph the slope is 4/1, or 4. 

The graph of a line of slope 2.

The equation in slope-intercept form is this:

y equals 4 x plus 2

Example 2

A line with a slope of 2 passes through the origin and through the point (k, 10). What is the value of k? 

A line that passes through the origin has a slope intercept of zero. This is the slope-intercept form.

The equation of a line of the form y = mx.

Now input the coordinates (k, 10) to find the value of k:

Solving the equation y=2x for y = 10.

Example 3

A line with a slope of -3 and y-intercept of 2 passes through the origin and through the point (k, 5). What is the value of k? 

Write the equation in slope-intercept form:

 

y equals negative 3 x plus 2

Now input the coordinates (k, 5) to find the value of k:

Solving the equation y = -3x+2 for y = 5.

Example 4

The line y = kx +3, where k is a constant, passes through the point (c, d). What is the slope of the line in terms of c and d?

In this case you’re given the equation in slope-intercept form. Simply plug in the values of c and d to find the value for k, which is the slope.

Solving the equation y=kx+3 for k.

Example 5

A line passes through the origin, as well as the points (3, k) and (k, 12). What is a possible value of k?

A line that passes through the origin has a slope intercept of zero. This is the slope-intercept form.

y=mx

Plug in the two sets of coordinates to get two separate equations:

Generating direct variations from two coordinates.

Now plug in the expression for k in the second expression and solve for m:

Solving 12=m•k.

There are two possible values for m, and, therefore, two separate values for k. This is one possible slope-intercept form with the one of the values for m:

y equals 2 x

Plug in one of the coordinates to solve for k:

Solving y=2x.


 

 

Slope-Based Questions

Some SAT questions rely on your understanding of slope. While the slope-intercept form includes the notion of slope, there are some additional concepts included.

 

First, let’s review the basics of slope. This is the slope formula:

 

Slope Formula. The equation used to find the slope m of a line, given two coordinates on the line.

Given two coordinates, the slope of the line connecting the two points is found using the slope formula.

On the SAT you may be asked about lines that are parallel or lines that are perpendicular. The key to understanding this is the slope-intercept form for parallel and perpendicular lines. Review these definitions:

 

Slopes of Parallel Lines. Lines that are parallel have the same slope.Slopes of Perpendicular Lines. Lines that are perpendicular lines have slopes that are negative reciprocals.

Basically, lines that are parallel have the same slope. Lines that are perpendicular have slopes that are negative reciprocals. Look at the following examples.

To see examples of finding the equations of parallel and perpendicular lines, click on the following links:

SAT Skill: Slope-Based Questions

These examples rely on a basic understanding of slope.

Example 1

The graph of a linear equation is 8x - 2y = 120. Decide if the graph of 3x + 5y = 150 is perpendicular to this line.

Write both equations in slope-intercept form:

Writing two equations in slope-intercept form to determine if they are perpendicular.

These slopes are not inverse reciprocals of each other. A line perpendicular to y = 4x - 60 would have a slope of -¼. A line perpendicular to y = -6x + 30 would have a slope of ⅙. These lines are not perpendicular to each other.

Example 2

The graph of a linear equation is -4x + 3y = 48. What is the equation of the line through (-1, 4) that is perpendicular to this line?

Write the equation in slope-intercept form: 

Writing the equation -4x+3y=48 in slope-intercept form.

A line perpendicular to this line would have a slope of -¾. Use the point-slope form to find the line that crosses (-1, 4):

Using the point-slope form to find an equation slope-intercept form.


 

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