Linear Equations in Two Variables
Find slopes and intercepts, write equations of lines, and test ordered pairs. Review parallel and perpendicular lines with worked SAT examples.
Linear Equations in Two Variables
SAT Math | Algebra | Foundation to Advanced
A linear equation in two variables describes a relationship between two quantities. On the SAT, you must understand how these equations represent straight lines, determine their slopes and intercepts, identify points on a line, write equations from given information, and interpret real-world situations.
1. Understanding Linear Equations in Two Variables
A linear equation in two variables is an equation involving two variables, usually x and y, that can be written in the form:
Ax + By = C
where A, B, and C are constants, and A and B cannot both be zero.
For example:
2x + 3y = 12
This equation contains two unknown quantities.
x is the first variable.
y is the second variable.
2 is the coefficient of x.
3 is the coefficient of y.
12 is the constant on the right side.
Unlike a typical linear equation in one variable, a linear equation in two variables generally has infinitely many ordered-pair solutions.
Each solution is written as an ordered pair:
(x, y)
For example, consider:
2x + 3y = 12
If x = 0:
3y = 12
y = 4
One solution is (0, 4).
If x = 3:
6 + 3y = 12
3y = 6
y = 2
Another solution is (3, 2).
If x = 6:
12 + 3y = 12
y = 0
Another solution is (6, 0).
All three points satisfy the same equation.
These points lie on a straight line in the coordinate plane.
Key idea: A linear equation in two variables represents a straight line containing all ordered pairs that satisfy the equation.
2. Understanding the Coordinate Plane
The coordinate plane contains two perpendicular axes.
The horizontal axis is called the x-axis.
The vertical axis is called the y-axis.
A point is written as:
(x, y)
The first coordinate shows the horizontal position.
The second coordinate shows the vertical position.
For example:
(3, 5) means move 3 units to the right and 5 units upward from the origin.
(−2, 4) means move 2 units to the left and 4 units upward.
The origin is the point:
(0, 0)
Checking whether a point lies on a line
To determine whether a point lies on a line, substitute both coordinates into the line's equation.
Worked Example 1 — Checking an ordered pair
Difficulty: Easy
Which of the following points lies on the line?
3x + 2y = 14
A. (1, 4)
B. (2, 4)
C. (3, 4)
D. (4, 4)
Solution
Substitute each ordered pair into the equation.
For A, (1, 4):
3(1) + 2(4) = 11
This is not 14.
For B, (2, 4):
3(2) + 2(4) = 6 + 8 = 14
The equation is satisfied.
For C, (3, 4):
3(3) + 2(4) = 17
For D, (4, 4):
3(4) + 2(4) = 20
Final answer: B. (2, 4)
SAT tip: Always substitute the first coordinate for x and the second coordinate for y.
3. Standard Form and Slope-Intercept Form
Linear equations can be written in different forms.
Two especially useful forms are:
Standard form:
Ax + By = C
Slope-intercept form:
y = mx + b
In slope-intercept form:
m represents the slope.
b represents the y-intercept.
Consider:
2x + y = 7
This is in standard form.
Subtract 2x from both sides:
y = −2x + 7
Now the equation is in slope-intercept form.
Therefore:
Slope = −2
Y-intercept = 7
Converting from standard form
For:
Ax + By = C
Subtract Ax:
By = −Ax + C
Divide by B, assuming B ≠ 0:
y = (−A/B)x + C/B
Therefore:
Slope = −A/B
Y-intercept = C/B
This shortcut only applies when the coefficient B is not zero.
Worked Example 2 — Converting equation forms
Difficulty: Medium
What is the slope of the line represented by:
6x + 3y = 15?
Step 1 — Isolate the y-term.
3y = −6x + 15
Step 2 — Divide every term by 3.
y = −2x + 5
Step 3 — Identify the slope.
The coefficient of x is −2.
Final answer: −2
SAT tip: The slope of Ax + By = C is −A/B, provided B is not zero. You do not always need to rewrite the entire equation.
4. Finding the X-Intercept and Y-Intercept
An intercept is a point where a line crosses an axis.
X-intercept
The x-intercept is where the line crosses the x-axis.
At the x-intercept:
y = 0
To find the x-intercept, substitute y = 0 into the equation and solve for x.
Y-intercept
The y-intercept is where the line crosses the y-axis.
At the y-intercept:
x = 0
To find the y-intercept, substitute x = 0 and solve for y.
Worked Example 3 — Finding both intercepts
Difficulty: Medium
Find the x-intercept and y-intercept of:
3x + 4y = 24
Part A — Find the x-intercept.
Set y = 0.
3x + 4(0) = 24
3x = 24
x = 8
The x-intercept is:
(8, 0)
Part B — Find the y-intercept.
Set x = 0.
3(0) + 4y = 24
4y = 24
y = 6
The y-intercept is:
(0, 6)
Final answers:
X-intercept: (8, 0)
Y-intercept: (0, 6)
SAT shortcut
For Ax + By = C, when A and B are both nonzero:
X-intercept = (C/A, 0)
Y-intercept = (0, C/B)
Remember that special cases exist, such as horizontal and vertical lines.
5. Understanding Slope
The slope describes a line's direction and steepness.
Slope is the ratio of the vertical change to the horizontal change between two different points.
Slope = Change in y / Change in x
The formula is:
m = (y₂ − y₁) / (x₂ − x₁)
The denominator must not equal zero.
Positive slope
A line with a positive slope rises from left to right.
Example:
y = 3x + 2
Slope = 3
Negative slope
A line with a negative slope falls from left to right.
Example:
y = −2x + 8
Slope = −2
Zero slope
A horizontal line has slope zero.
Example:
y = 5
Undefined slope
A vertical line has undefined slope.
Example:
x = 4
Worked Example 4 — Finding slope from two points
Difficulty: Medium
A line passes through:
(−2, 7) and (4, −5)
What is its slope?
Step 1 — Write the slope formula.
m = (y₂ − y₁) / (x₂ − x₁)
Step 2 — Substitute the coordinates.
m = (−5 − 7) / (4 − (−2))
Step 3 — Simplify.
m = −12 / 6
m = −2
Final answer: −2
SAT tip: Keep the subtraction order consistent. Subtract the coordinates of the first point from the corresponding coordinates of the second point.
6. Writing the Equation of a Line
The SAT may give you two points, a slope and a point, or information about intercepts.
You must use that information to determine the line's equation.
Method 1 — Given slope and y-intercept
Use:
y = mx + b
Example:
Slope = 4
Y-intercept = −3
The equation is:
y = 4x − 3
Method 2 — Given slope and one point
If the slope and a point (x₁, y₁) are given, use the point-slope formula:
y − y₁ = m(x − x₁)
You can then simplify into slope-intercept or standard form.
Worked Example 5 — Slope and a point
Difficulty: Medium
A line has slope 3 and passes through the point (2, 5).
What is the equation of the line?
Step 1 — Use point-slope form.
y − 5 = 3(x − 2)
Step 2 — Distribute.
y − 5 = 3x − 6
Step 3 — Add 5.
y = 3x − 1
Final answer: y = 3x − 1
Method 3 — Given two points
When two points are given:
Calculate the slope.
Substitute the slope into y = mx + b.
Use either point to determine b.
Write the complete equation.
Worked Example 6 — Writing an equation from two points
Difficulty: Medium
Which equation represents the line through:
(1, 4) and (3, 10)?
Step 1 — Find the slope.
m = (10 − 4) / (3 − 1)
m = 6/2
m = 3
Step 2 — Write the general equation.
y = 3x + b
Step 3 — Substitute (1, 4).
4 = 3(1) + b
4 = 3 + b
b = 1
Step 4 — Write the equation.
y = 3x + 1
Check using the second point:
10 = 3(3) + 1
10 = 10
Final answer: y = 3x + 1
7. Parallel Lines
Parallel lines are distinct lines in the same plane that never intersect.
For two nonvertical lines to be parallel, they must have the same slope.
For example:
y = 2x + 3
y = 2x − 5
Both lines have slope 2.
Their y-intercepts are different.
Therefore, the lines are parallel.
Important rule
For nonvertical lines:
Parallel lines have equal slopes.
Two distinct vertical lines are also parallel.
Worked Example 7 — Parallel line through a point
Difficulty: Medium–Hard
A line is parallel to:
y = −3x + 8
and passes through:
(2, 5)
Which equation represents this line?
Step 1 — Identify the slope.
The given line has slope −3.
A parallel line must also have slope −3.
Step 2 — Write the new equation.
y = −3x + b
Step 3 — Substitute the point.
5 = −3(2) + b
5 = −6 + b
b = 11
Step 4 — Write the equation.
y = −3x + 11
Final answer: y = −3x + 11
SAT tip: Parallel lines have equal slopes, but their intercepts may differ.
8. Perpendicular Lines
Perpendicular lines intersect at a right angle of 90°.
For two nonvertical, nonhorizontal perpendicular lines, their slopes are negative reciprocals.
If one slope is m, the perpendicular slope is:
−1/m
For example:
If one line has slope 2, a perpendicular line has slope −1/2.
If one line has slope −3, a perpendicular line has slope 1/3.
If one line has slope −2/5, a perpendicular line has slope 5/2.
Important rule
For two perpendicular lines with defined, nonzero slopes:
m₁ × m₂ = −1
A horizontal line and a vertical line are also perpendicular.
Worked Example 8 — Perpendicular line through a point
Difficulty: Hard
A line is perpendicular to:
y = 2x + 1
and passes through:
(4, −1)
What is the equation of the line?
Step 1 — Identify the original slope.
m = 2
Step 2 — Find the negative reciprocal.
New slope = −1/2
Step 3 — Write the equation.
y = −(1/2)x + b
Step 4 — Substitute (4, −1).
−1 = −(1/2)(4) + b
−1 = −2 + b
b = 1
Step 5 — Write the final equation.
y = −(1/2)x + 1
Final answer: y = −(1/2)x + 1
Common mistake: Changing only the sign of the slope does not always produce a perpendicular line. You must also take its reciprocal.
9. Horizontal and Vertical Lines
Horizontal and vertical lines are special cases.
Horizontal lines
A horizontal line has the form:
y = c
where c is a constant.
Example:
y = 6
Every point on this line has a y-coordinate of 6.
Examples of points on this line:
(0, 6), (2, 6), (−5, 6)
The slope of a horizontal line is zero.
Vertical lines
A vertical line has the form:
x = c
where c is a constant.
Example:
x = −4
Every point on this line has an x-coordinate of −4.
Examples:
(−4, 0), (−4, 3), (−4, −6)
The slope of a vertical line is undefined.
A vertical line is not a function of x because one x-value is associated with many y-values.
SAT tip: Do not confuse x = 4 with y = 4.
x = 4 is vertical.
y = 4 is horizontal.
10. Connecting Equations, Tables, and Graphs
A linear relationship can be represented using an equation, a table, or a graph.
These representations describe the same mathematical relationship in different ways.
Consider:
y = 2x + 3
A corresponding table is:
x | y |
|---|---|
0 | 3 |
1 | 5 |
2 | 7 |
3 | 9 |
The graph is the straight line passing through these points.
Turning a table into an equation
Worked Example 9 — Finding the equation from a table
Difficulty: Medium
A line contains the following points:
x | y |
|---|---|
0 | 7 |
2 | 1 |
4 | −5 |
What is its equation?
Step 1 — Find the slope.
m = (1 − 7) / (2 − 0)
m = −6/2
m = −3
Step 2 — Identify the intercept.
When x = 0, y = 7.
Therefore b = 7.
Step 3 — Write the equation.
y = −3x + 7
Final answer: y = −3x + 7
Graphing from intercepts
For the line:
2x + 3y = 12
Set x = 0:
3y = 12
y = 4
The y-intercept is (0, 4).
Set y = 0:
2x = 12
x = 6
The x-intercept is (6, 0).
Plot these two points and draw a straight line through them.
Two distinct points are enough to determine a unique straight line.
11. Creating Equations from Real-World Situations
Linear equations in two variables can describe constraints involving two different quantities.
For example, consider two types of tickets sold at different prices.
If adult tickets cost $12 each and student tickets cost $7 each, the total revenue is:
12a + 7s = R
where:
a is the number of adult tickets.
s is the number of student tickets.
R is the total revenue.
This equation connects two quantities without requiring one to be the input and the other to be the output.
Worked Example 10 — Ticket sales
Difficulty: Medium
A theater sells adult tickets for $12 and student tickets for $7.
The total revenue is $310.
If 13 adult tickets are sold, how many student tickets are sold?
Step 1 — Define the variables.
a = number of adult tickets
s = number of student tickets
Step 2 — Write the revenue equation.
12a + 7s = 310
Step 3 — Substitute a = 13.
12(13) + 7s = 310
156 + 7s = 310
Step 4 — Subtract 156.
7s = 154
Step 5 — Divide by 7.
s = 22
Final answer: 22 student tickets
Interpreting solutions in context
Mathematically, a line may contain infinitely many ordered pairs.
However, some real-world contexts restrict which solutions make sense.
For example, ticket counts cannot be negative or fractional.
If a problem concerns numbers of tickets, the relevant solutions must use nonnegative whole numbers.
SAT tip: Always check whether your answer makes sense in the real-world situation.
12. Advanced SAT Questions with Unknown Constants
Some difficult SAT questions include unknown coefficients or constants.
Instead of asking you to solve for x or y directly, these questions may ask for an unknown number that makes a line satisfy a given condition.
Worked Example 11 — Unknown coefficient
Difficulty: Hard
The equation:
kx + 4y = 16
represents a line with slope −3/2.
What is k?
Step 1 — Rewrite in slope-intercept form.
4y = −kx + 16
Divide by 4:
y = −(k/4)x + 4
Step 2 — Identify the slope.
Slope = −k/4
Step 3 — Set the slope equal to −3/2.
−k/4 = −3/2
Multiply by −4:
k = 6
Final answer: 6
Worked Example 12 — Unknown coordinate
Difficulty: Hard
The line passes through:
(3, −2) and (k, 10)
Its slope is 4.
What is the value of k?
Step 1 — Apply the slope formula.
4 = (10 − (−2)) / (k − 3)
Step 2 — Simplify.
4 = 12 / (k − 3)
Step 3 — Multiply both sides.
4(k − 3) = 12
Step 4 — Divide by 4.
k − 3 = 3
Step 5 — Add 3.
k = 6
Final answer: 6
Equivalent equations of the same line
Two equations can represent the same line even if their coefficients look different.
For example:
2x − 3y = 12
Multiply every term by 2:
4x − 6y = 24
These equations represent exactly the same line.
Important: To preserve the same equation, multiply or divide every term by the same nonzero number.
13. SAT Strategies and Common Mistakes
Strategy 1 — Choose the fastest representation
If a line is already written as y = mx + b, read the slope and intercept directly.
If the line is given in standard form, use −A/B for the slope when B ≠ 0.
If two points are given, calculate the slope using the slope formula.
Strategy 2 — Use substitution to check points
If the question asks whether an ordered pair lies on a line, substitute its coordinates into the equation.
This is often faster than drawing a graph.
Strategy 3 — Remember the intercept rules
For the x-intercept, set y = 0.
For the y-intercept, set x = 0.
Strategy 4 — Recognize parallel and perpendicular lines
Parallel nonvertical lines have equal slopes.
Perpendicular lines with defined nonzero slopes have negative reciprocal slopes.
Strategy 5 — Simplify equations carefully
When converting standard form into slope-intercept form, divide every term by the coefficient of y.
Strategy 6 — Use Desmos strategically
The Digital SAT provides an embedded Desmos calculator.
You can use it to visualize equations, inspect intercepts, or compare graphs.
However, simple algebraic methods may be faster than graphing every question.
Common Mistake 1 — Confusing x and y
In the ordered pair (4, 7), x = 4 and y = 7.
Do not reverse the coordinates during substitution.
Common Mistake 2 — Reversing slope calculations
Correct:
Slope = Change in y / Change in x
Incorrect:
Slope = Change in x / Change in y
Common Mistake 3 — Confusing intercepts
The x-intercept has y = 0.
The y-intercept has x = 0.
Common Mistake 4 — Losing a negative sign
For:
3x + 2y = 12
Subtract 3x:
2y = −3x + 12
The slope is negative:
−3/2
Common Mistake 5 — Assuming parallel lines have the same intercept
Parallel lines must have the same slope, but distinct parallel lines have different y-intercepts.
Common Mistake 6 — Finding the reciprocal but forgetting the negative sign
A slope of 2 has perpendicular slope −1/2, not 1/2.
Common Mistake 7 — Treating vertical and horizontal lines the same
y = 5 is horizontal with slope 0.
x = 5 is vertical with undefined slope.
Common Mistake 8 — Ignoring context
A mathematical solution involving a negative number of tickets or a fraction of a student may be invalid in the context of the problem.
Common Mistake 9 — Changing only part of an equation
When multiplying an equation by a constant, multiply every term on both sides by that constant.
Common Mistake 10 — Solving more than necessary
If a problem gives a line's slope and asks for the slope of a parallel line, the answer is already known.
You do not need to determine the full equation unless requested.
14. Check Your Understanding
Question 1 — Ordered pairs
Which point lies on:
2x + y = 11?
A. (2, 5)
B. (3, 5)
C. (4, 5)
D. (5, 5)
Correct answer: B
Substitute (3, 5):
2(3) + 5 = 11
The equation is satisfied.
Question 2 — Finding slope
What is the slope of:
4x + 2y = 10?
A. −4
B. −2
C. 2
D. 4
Correct answer: B
Rearrange:
2y = −4x + 10
y = −2x + 5
The slope is −2.
Question 3 — Parallel lines
Which line is parallel to y = 3x − 7?
A. y = −3x + 2
B. y = (1/3)x + 2
C. y = 3x + 5
D. y = −(1/3)x + 5
Correct answer: C
Parallel nonvertical lines have equal slopes.
Both lines have slope 3.
Question 4 — Intercepts
What is the x-intercept of:
5x + 2y = 30?
A. 5
B. 6
C. 10
D. 15
Correct answer: B
Set y = 0:
5x = 30
x = 6
Question 5 — Perpendicular lines
A line has slope −4.
What is the slope of a perpendicular line?
A. −1/4
B. 1/4
C. 4
D. −4
Correct answer: B
The negative reciprocal of −4 is 1/4.
15. Lesson Summary
A linear equation in two variables represents a straight line in the coordinate plane.
The general form is:
Ax + By = C
A solution is an ordered pair (x, y) that satisfies the equation.
The slope-intercept form is:
y = mx + b
The slope describes the direction and steepness of the line.
The y-intercept is where the line crosses the y-axis.
The x-intercept is where the line crosses the x-axis.
Two distinct points determine a unique straight line.
Parallel nonvertical lines have the same slope.
Perpendicular nonvertical, nonhorizontal lines have negative reciprocal slopes.
Horizontal lines have slope zero.
Vertical lines have undefined slope.
Linear equations can model real-world constraints involving two quantities, such as ticket prices, purchases, distances, and rates.
You are now ready to complete the Linear Equations in Two Variables practice bank in Baloo Prep.