Why Linear Equations Matter on the SAT®

Linear relationships appear in nearly \boldsymbol{25\%} of SAT® Math questions. Therefore, a solid grip on them, including two variable equations​, can raise a score quickly. Each question is really a small puzzle: turn an everyday situation into a solvable rule, then read the rule for answers. This guide builds those practical skills step by step.

What Is a Linear Equation in Two Variables?

A linear equation in two variables is a rule that links two quantities (usually x and y) in a straight-line pattern.

Three standard forms dominate the SAT®:

• Slope–intercept form: y = mx + b
• Standard form: Ax + By = C
• Point–slope form: y - y_1 = m(x - x_1)

Because the equation is first-degree in both variables, its graph is always a straight line.

Slope–Intercept Form: y = mx + b

Breaking Down the Parts

1. Slope (m) – change in y for every 1-unit change in x.
2. y-intercept (b) – starting value when x = 0.

Worked Example

A babysitter charges a \$12 travel fee plus \$9 per hour. Write an equation for total cost (C) after h hours, then find the cost of 4 hours.

Step 1: Define the variables
Let h represent the number of hours the babysitter works, and let C represent the total cost.

Step 2: Identify the rate (slope)
The babysitter charges \$9 per hour, so the rate is 9.

Step 3: Identify the starting value (intercept)
There is a one-time travel fee of \$12, which is the initial fixed cost.

Step 4: Write the equation
Using the slope-intercept form C = 9h + 12, where 9 is the rate per hour and \$12 is the fixed travel fee.

Step 5: Substitute h = 4 to find the cost for 4 hours
C = 9(4) + 12 = 36 + 12 = 48

The total cost for 4 hours is \$48.

Quick Tip! When a question shows numbers like Ax + By = C, rewrite to y = mx + b to read slope and intercept instantly.

Creating Two-Variable Equations From Word Problems

Four-Step Strategy

1. Identify the two quantities.
2. Choose variables.
3. Locate slope (rate) and intercept (starting amount).
4. Write and check the equation.

Real-World Example

A phone plan costs \$20 per month plus \$3 for each gigabyte of data used.

Step 1: Identify the quantities
The two quantities involved are data usage in gigabytes (g) and total monthly cost (C).

Step 2: Define the variables
Let g represent the number of gigabytes used and C represent the total monthly cost.

Step 3: Determine the rate and starting value
The rate (slope) is \$3 per gigabyte, and the starting value (intercept) is the fixed monthly fee of \$20.

Step 4: Write the equation
Using slope-intercept form, the equation is C = 3g + 20.

Step 5: Substitute g = 7 to find the monthly cost
C = 3(7) + 20 = 21 + 20 = 41

If 7 GB are used, the total monthly bill is \$41.

Interpreting Parts of a Two-Variable Equation in Context

Once a model is set, each number tells a story.

The equation 4x + 2y = 40 models a snack budget where x represents the number of pretzels (\$4 each) and y represents the number of drinks (\$2 each).

• Coefficient 4 → cost per pretzel.
• Coefficient 2 → cost per drink.
• Constant 40 → total money available.

Seeing the structure can help. Let’s divide both sides by 2 → 2x + y = 20

Now we can see that the cost per drink is \$1, which may speed mental math.

Multiple Representations of Two-Variable Equations

The SAT® often toggles among three representations of two-variable equations: tables, graphs, and equations.

Example: y = -2x + 6

Here are some tips to change from one representation to another.

1. Equation → Table
   • Pick x-values (–1, 0, 1, 2). Compute y.
2. Table → Graph
   • Plot the points found above.
3. Graph → Equation
   • \text{Slope}=\frac{\text{Rise}}{\text{Run}}=–2
   • y-intercept is 6 → equation must be y = -2x + 6.

Finding Missing Values in Two-Variable Equations

If one value in a two-variable equation is known, we can substitute the known variable and solve for the other.

Example: Given 5x - 3y = 12, find y when x = 3.

Step	Reason
5(3) - 3y = 12	Substitute x=3 into the equation
15 - 3y = 12	Simplify
15 \color{red}{-15} -3y = 12 \color{red}{-15}	Subtract 15 from both sides
-3y = -3	Simplify
\dfrac{-3y}{\color{red}{-3}} = \dfrac{-3}{\color{red}{-3}}	Divide both sides by -3
y = 1	Final answer ✅

Therefore, the ordered pair is (3, 1).

Writing a Two-Variable Equation When You Have…

Two Points

Given (–2, 5) and (4, –1).

• Slope: m = \dfrac{-1 - 5}{4 - (-2)} = \dfrac{-6}{6} = -1
• Use one point in point-slope: y - 5 = -1(x + 2)
• Simplify → y = -x + 3

One Point + Slope

Point (1, 8), slope 3.

Write the equation in Point-slope form and convert to slope-intercept form, if needed.

y - 8 = 3(x - 1) → y = 3x + 5

One Point + Parallel Line

Through (–3, 4), parallel to 2x - y = 7.

• Parallel → same slope. First, solve the original equation for y:
  • -y = -2x + 7 \Rightarrow y = 2x - 7; slope = 2.
• Now: y - 4 = 2(x + 3) → y = 2x + 10

One Point + Perpendicular Line

Through (2, 5), perpendicular to y = -\frac{1}{2}x + 6.

• Perpendicular slope → negative reciprocal of –\frac{1}{2} = 2.
• Equation: y - 5 = 2(x - 2) → y = 2x + 1

Common Pitfalls and Time-Saving SAT® Tips

• Sign slips with negative slopes—slow down on subtraction.
• Intercept mix-ups—remember the x-intercept occurs when y = 0 and the y-intercept occurs when x=0
• Calculator table trick—On graphing calculators, “TABLE” displays (x, y) quickly; use it to hunt integer solutions.
• 30-second elimination—If only one answer choice has the correct y-intercept, pick it without further work.

Two-Variable Equations Quick Reference Chart

Term	Student-Friendly Definition
Slope (m)	How much y changes for a 1-unit change in x
y-intercept (b)	Point where the line crosses the y-axis (x = 0)
Standard Form	Ax + By = C where A, B, C are integers
Parallel Lines	Lines with equal slopes, never meeting
Perpendicular Lines	Lines whose slopes multiply to –1
Rate of Change	Another name for slope in context
Constant Term	The standalone number (e.g., b)
Solution Point	Any (x, y) making the equation true
x-Intercept	Point where y = 0
Linear Equation in Two Variables	First-degree equation relating x and y

Two-Variable Equation Practice Problems

1. A gym charges \$15 sign-up plus \$25 per month.
   • a) Write an equation for total cost (C) after m months.
   • b) How much after 6 months?
2. Find the equation of the line through (–4, 2) and (2, –3).
3. A line has the equation 3x + 2y = 18.
   • a) What is its slope?
   • b) What is the y-intercept?
4. Write an equation of the line perpendicular to y = 4x - 9 that passes through (0, –2).

Answers

1. a) C = 25m + 15; b) \$165
2. y = -\tfrac{5}{6}x - \tfrac{1}{3}
3. a) slope = –1.5; b) y-intercept = 9
4. y = -\frac{1}{4}x - 2

Next Steps for Mastery

Mastery grows with repeated, spaced practice. Therefore, revisit these steps while studying systems of equations and linear inequalities; the ideas overlap. Finally, timed quizzes and official SAT® practice tests will lock in speed and confidence. Keep building—each new line written is a point earned on test day.

Sharpen Your Skills for SAT® Math (Digital)

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