Linear Equations | SAT Algebra | SAT Math

Introduction

Linear equations form a crucial part of the SAT math reasoning section, primarily emphasizing algebraic understanding and problem-solving skills. These questions require the manipulation of equations to solve for unknown variables and, sometimes, to interpret the relationships they describe.

Approach Question

What is the slope of the line in the (x,y)-coordinate plane that passes through the point (-4, 4) and has a y-intercept of -8?

A. -2
B. -3
C. -4
D. 4

Explanation

The slope of a line in the coordinate plane is calculated using the following formula:

$slope = \frac{Δ y}{Δ x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Here we’re referring to changes (specifically, differences or the results of subtraction) in the y and x coordinates, respectively. Given the point (-4, 4) and the y-intercept (0, -8), we can calculate the slope as follows:

Change in $y = 4 - (- 8) = 12$

Change in $x = - 4 - 0 = - 4$

$slope = Δ y /Δ x = 12/ - 4 = - 3$

The answer is B.

This slope indicates that for every 1 unit the line moves horizontally to the right, it moves 3 units vertically down.

Definitions

Linear Equation: An equation involving only first-degree polynomials. For example, $a x + b = c$ is a linear equation in one variable ( $a$ , $b$ , and $c$ would be constants in this case).
Variable: A symbol used to represent a number in equations and expressions.
Constant: A value that does not change.
Slope: A measure of the steepness of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on the line.

Topics for Cross-Reference

Modeling Equations and Inequalities
Systems
Linear Inequalities

Variations

The majority of linear equations questions ask you to model a real-life situation using a linear equation (or interpret an existing linear equation constructed to model a real-life situation). This task is so frequent that we have included a separate module to cover the topic. This module and associated practice questions focus on pure linear equations without a real-life connection.

Strategy Insights

Always isolate the variable to one side of the equation to simplify the solving process.
Remember to perform any operation on both sides of an equation to maintain equality.
Check your solutions by substituting them back into the original equation to ensure they satisfy the equation.
Watch out for reflection questions, where a line is reflected over either the x-axis or the y-axis. The rule of thumb here is: when a line is reflected over an axis, you negate the opposite variable. (So, if we are reflecting over the x-axis, the y-coordinate gets negated while the x-coordinate stays the same.)

Flashcard Fodder

Finding slope from two points: $\frac{r i se}{r u n}$ = $\frac{Δ y}{Δ x}$ = $\frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Standard form of a linear equation: $A x + B y = C$
The slope of a line in standard form: $- \frac{A}{B}$
Slope-intercept form of a linear equation: $y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept
Parallel lines have identical slopes.
Perpendicular lines have negative reciprocal slopes.

Sample Questions

Difficulty 1

A line in the xy-plane has a slope of $\frac{1}{5}$ and passes through the point $(0, - 4)$ . Which equation represents this line?

A. $y = \frac{1}{5} x + 4$
B. $y = \frac{1}{5} x - 4$
C. $y = - \frac{1}{5} x + 4$
D. $y = - \frac{1}{5} x - 4$

Take a minute to solve, then check your answer.

(spoiler)

The answer is $y = \frac{1}{5} x - 4$ . To answer this question, you must be able to 1) recognize slope-intercept form ( $y = m x + b$ ), and 2) recognize that this question already provides the $y$ -intercept, since the $y$ -intercept occurs whenever $x = 0$ .

With this in mind, we look for a slope of $\frac{1}{5}$ and a $y$ -intercept of $- 4$ . The answer of $y = \frac{1}{5} x - 4$ lines up.

Difficulty 2

Line $S$ is parallel to a line with equation $2 x + 4 y = 8$ and has an x-intercept of 4. What is the $y$ -intercept of line $S$ ?

A. 0
B. 2
C. 4
D. 6

(spoiler)

The answer is 2. The line is presented to us in standard form ( $A x + B y = C$ ). As noted under Flashcard Fodder for this section, the slope of a line in standard form is equal to $- \frac{A}{B}$ . Why is this helpful? Because, as also noted in this lesson, parallel lines have the same slope. So not only does the original line have a slope of $- \frac{2}{4} = - \frac{1}{2}$ , but Line $S$ , we know, does as well.

How, then, do we use the information about the $x$ -intercept? The key principle here is that you can always plug a given point into the equation of the line on which that point is found. If the $x$ -intercept is $4$ , then the line must pass through the point $(4, 0)$ . Framing the slope-intercept form of Line $S$ as $y = - \frac{1}{2} + b$ and plugging in our point, we get $0 = - \frac{1}{2} (4) + b$ . Solving for $b$ by simplifying and adding $2$ to both sides, we come up with $b = 2$ . That’s our $y$ -intercept!

Difficulty 3

Line $j$ is defined by $5 x + 4 y = 17$ . Line $p$ is perpendicular to line $j$ in the xy-plane. What is the slope of line $p$ ? (Note: this is a free-response question.)

(spoiler)

The answer is $\frac{4}{5}$ . Many, if not most, students, will determine the slope of line $j$ by manipulating the equation into slope-intercept form and reading the $m$ , but there is a quicker way: the slope of a line in standard form is equal to $- \frac{A}{B}$ . Applying this formula, we arrive quickly at $- \frac{5}{4}$ for the slope of line $j$ .

How does this help us find the slope of line $p$ ? Remember that perpendicular lines have negative reciprocal slopes. So we need to flip and negate the fraction $- \frac{5}{4}$ , giving us $\frac{4}{5}$ .

Difficulty 4

In the xy-plane, line $s$ passes through the point $(0, 0)$ and is parallel to the line given by $2 x - 7 y = 21$ . If line $s$ also passes through the point $(14, c)$ , what is the value of $c$ ?

A. -7
B. -4
C. 4
D. 7

(spoiler)

The answer is 4. Parallel lines are simpler than perpendicular lines in that parallel lines simply have the same slope as each other. S if we find the slope of the line given by the equation, we will acquire the slope of line $s$ as well. As already shown in this lesson, the best way to find the slope of a line in standard form is $- \frac{A}{B}$ . Applying that formula, we learn that the slope of both lines in this problem is $\frac{2}{7}$ .

Once you find the slope, the next step is often to use slope-intercept form in some way. The purpose of this step is often to find the $y$ -intercept, but if you look carefully, you will see that we already know the $y$ -intercept in this case: if the line passes through $(0, 0)$ , then the $y$ -intercept is zero! Since we know the $y$ -intercept, we know that the slope-intercept form of line $s$ is $y = \frac{2}{7} x$ .

Now that we know the slope-intercept form equations, we can make use of one of the most common linear equation tools: plugging in a known point in order to solve for a variable. In this case, though, only the $x$ -coordinate of $14$ is known; our task is to solve for the $y$ -coordinate, represented in this problem by the variable $c$ . Plugging in our point into our known equation, we find:

$y c c c = \frac{2}{7} x = \frac{2}{7} (14) = \frac{28}{7} = 4$

Difficulty 5

The graph of the equation $8 x + 7 y = 33$ is translated up three units in the $x y$ -plane. What is the $x$ -coordinate of the $x$ -intercept of the resulting graph? (Note: this is a free-response question.)

(spoiler)

The answer is $\frac{27}{4}$ . The translation in this question is deceptively challenging because we cannot simply add $3$ to $33$ to move up $3$ units. This is because, in standard form, $C$ does not represent a fixed coordinate as $b$ does in slope-intercept form. What then? Let’s convert the equation to slope-intercept form.

$8 x + 7 y 7 y y = 33 = - 8 x + 33 = - \frac{8}{7} x + \frac{33}{7}$

Now we can add three to the constant in our equation to make a true translation. Using the common denominator process for adding fractions, we find the translated equation is $y = - \frac{8}{7} x + \frac{54}{7}$ . How do we find the $x$ -intercept? Remember that an intercept occurs when the opposite coordinate equals zero, so we need to set $y$ equal to zero and solve for $x$ . Here’s what results:

$y 0 - \frac{54}{7} x = - \frac{8}{7} x + \frac{54}{7} = - \frac{8}{7} x + \frac{54}{7} = - \frac{8}{7} x = \frac{54}{8} = \frac{27}{4}$

For Reflection

What strategies do you find most effective when solving linear equations?
Rate the difficulty of linear equations for you from 1 (easy) to 5 (challenging). How comfortable are you with manipulating equations?
Have you ever applied linear equations to real-world problems? If so, describe how understanding linear equations helped you do so.

Introduction

Approach Question

What is the slope of the line in the (x,y)-coordinate plane that passes through the point (-4, 4) and has a y-intercept of -8?

A. -2
B. -3
C. -4
D. 4

Explanation

The slope of a line in the coordinate plane is calculated using the following formula:

$slope = \frac{Δ y}{Δ x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Change in $y = 4 - (- 8) = 12$

Change in $x = - 4 - 0 = - 4$

$slope = Δ y /Δ x = 12/ - 4 = - 3$

The answer is B.

This slope indicates that for every 1 unit the line moves horizontally to the right, it moves 3 units vertically down.

Definitions

Linear Equation: An equation involving only first-degree polynomials. For example, $a x + b = c$ is a linear equation in one variable ( $a$ , $b$ , and $c$ would be constants in this case).
Variable: A symbol used to represent a number in equations and expressions.
Constant: A value that does not change.
Slope: A measure of the steepness of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on the line.

Topics for Cross-Reference

Modeling Equations and Inequalities
Systems
Linear Inequalities

Variations

Strategy Insights

Always isolate the variable to one side of the equation to simplify the solving process.
Remember to perform any operation on both sides of an equation to maintain equality.
Check your solutions by substituting them back into the original equation to ensure they satisfy the equation.
Watch out for reflection questions, where a line is reflected over either the x-axis or the y-axis. The rule of thumb here is: when a line is reflected over an axis, you negate the opposite variable. (So, if we are reflecting over the x-axis, the y-coordinate gets negated while the x-coordinate stays the same.)

Flashcard Fodder

Finding slope from two points: $\frac{r i se}{r u n}$ = $\frac{Δ y}{Δ x}$ = $\frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Standard form of a linear equation: $A x + B y = C$
The slope of a line in standard form: $- \frac{A}{B}$
Slope-intercept form of a linear equation: $y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept
Parallel lines have identical slopes.
Perpendicular lines have negative reciprocal slopes.

Sample Questions

Difficulty 1

A line in the xy-plane has a slope of $\frac{1}{5}$ and passes through the point $(0, - 4)$ . Which equation represents this line?

A. $y = \frac{1}{5} x + 4$
B. $y = \frac{1}{5} x - 4$
C. $y = - \frac{1}{5} x + 4$
D. $y = - \frac{1}{5} x - 4$

Take a minute to solve, then check your answer.

(spoiler)

With this in mind, we look for a slope of $\frac{1}{5}$ and a $y$ -intercept of $- 4$ . The answer of $y = \frac{1}{5} x - 4$ lines up.

Difficulty 2

Line $S$ is parallel to a line with equation $2 x + 4 y = 8$ and has an x-intercept of 4. What is the $y$ -intercept of line $S$ ?

A. 0
B. 2
C. 4
D. 6

(spoiler)

Difficulty 3

Line $j$ is defined by $5 x + 4 y = 17$ . Line $p$ is perpendicular to line $j$ in the xy-plane. What is the slope of line $p$ ? (Note: this is a free-response question.)

(spoiler)

Difficulty 4

In the xy-plane, line $s$ passes through the point $(0, 0)$ and is parallel to the line given by $2 x - 7 y = 21$ . If line $s$ also passes through the point $(14, c)$ , what is the value of $c$ ?

A. -7
B. -4
C. 4
D. 7

(spoiler)

$y c c c = \frac{2}{7} x = \frac{2}{7} (14) = \frac{28}{7} = 4$

Difficulty 5

The graph of the equation $8 x + 7 y = 33$ is translated up three units in the $x y$ -plane. What is the $x$ -coordinate of the $x$ -intercept of the resulting graph? (Note: this is a free-response question.)

(spoiler)

$8 x + 7 y 7 y y = 33 = - 8 x + 33 = - \frac{8}{7} x + \frac{33}{7}$

$y 0 - \frac{54}{7} x = - \frac{8}{7} x + \frac{54}{7} = - \frac{8}{7} x + \frac{54}{7} = - \frac{8}{7} x = \frac{54}{8} = \frac{27}{4}$

For Reflection

What strategies do you find most effective when solving linear equations?
Rate the difficulty of linear equations for you from 1 (easy) to 5 (challenging). How comfortable are you with manipulating equations?
Have you ever applied linear equations to real-world problems? If so, describe how understanding linear equations helped you do so.