FREE SAT STUDY GUIDE
SAT Algebra
Learn linear equations, functions, inequalities, and systems with free SAT algebra lessons and worked examples.
These are BuildSAT teaching examples. Use the study workspace to apply the concepts to official practice questions and review your progress.
Open interactive lessons →Linear equations in one variable
Translate words into expressions
Turn a situation into an equation with meaningful units.
Define the unknown
Write what the variable measures, including its unit. A fixed charge is added once; a per-item charge is multiplied by the item count.
Group operations
“Three times the sum of x and 4” is 3(x + 4), while “four more than three times x” is 3x + 4. Distribute multiplication to every term inside parentheses.
An expression describes a quantity; an equation asserts equality
A variable stands for a number that may be unknown or may change. A coefficient multiplies a variable; a constant does not change with it. Thus 5n + 8 describes a quantity, while 5n + 8 = 33 states that the quantity equals 33. Define the variable before deciding what relationship to write.
Build the model from units and repeated groups
A rate of $5 per item multiplied by n items gives 5n dollars because the item units cancel. Add a one-time $8 charge only once. Parentheses tell you which operations form a group: “twice the total” multiplies the entire total, not just its first term.
Worked example
A bike rental costs $6 plus $4 per hour. Write the cost for h hours.
- The $6 fee is fixed.
- The variable cost is 4h dollars.
C = 6 + 4h
Watch for this: “Less than” can reverse the word order: 5 less than x is x − 5.
Check your understanding
A club charges $12 to join and $7 per class. What is the cost of c classes?
- 19c
- 12c + 7
- 12 + 7c
- 7(c + 12)
Show answer and explanation
Answer: C. The one-time fee is 12 and the class charge is 7c, giving 12 + 7c.
Linear equations in one variable
Solve linear equations
Isolate a variable and recognize special cases.
Keep the sides equal
Distribute, combine like terms, and perform the same operation on both sides. Undo addition before multiplication when isolating a simple variable term.
No solution or infinitely many
If the variable cancels and leaves a false statement such as 2 = 5, there is no solution. A true identity such as 2 = 2 means every permitted value works.
Equality behaves like a balanced scale
An equation states that two expressions have equal values. Adding the same amount to both sides or multiplying both by the same nonzero number preserves that equality. Solving reverses the operations around the variable until its value is isolated. Each step should produce an equivalent equation with the same solutions.
A linear equation does not always have exactly one answer
After simplification, ax + b = cx + d becomes (a − c)x = d − b. A nonzero coefficient gives one solution. If the coefficient is zero, inspect the constants: 0 = 0 is true for every permitted input, while 0 = 5 is false for every input.
Worked example
Solve 3(x − 2) + 4 = 16.
- Distribute: 3x − 6 + 4 = 16.
- Combine and add 2: 3x = 18.
- Divide by 3 and substitute to check.
x = 6
Watch for this: The question may ask for 4x, x + 2, or another expression rather than x itself.
Check your understanding
What is the solution to 4x + 9 = 25?
- 4
- 8
- 16
- 34
Show answer and explanation
Answer: A. Subtract 9 to get 4x = 16; divide by 4 to get x = 4.
Linear functions
Slope, intercept, and linear models
Interpret the rate of change and starting value of a linear function.
Slope-intercept form
In y = mx + b, m is the change in y per one-unit increase in x, and b is y when x = 0. Slope is (y₂ − y₁)/(x₂ − x₁).
Interpret the context
A negative slope represents a decrease. An intercept may be a starting amount, but only if x = 0 has meaning in the situation.
A function pairs each allowed input with one output
A linear function changes its output by the same amount whenever the input increases by the same amount. This constant rate is its slope. In f(x) = mx + b, the term mx measures accumulated change from x = 0 and b gives the output at that starting input.
Slope carries units and a sign
If distance in kilometers is plotted against time in hours, slope has units kilometers per hour. Compute the change in output divided by the matching change in input, keeping the subtraction order consistent. A negative slope indicates a decrease in output as input increases, not a negative time interval.
Worked example
A tank contains W(t) = 90 − 3t liters after t minutes.
- At t = 0, W = 90 liters.
- Each additional minute reduces the volume by 3 liters.
Starting volume: 90 L; rate: −3 L/min.
Watch for this: Use change in output divided by change in input, not the reverse.
Check your understanding
In C(n) = 18 + 5n, C is the dollar cost of n tickets. What does 5 represent?
- The initial fee
- The number of tickets
- The maximum cost
- The dollar cost per additional ticket
Show answer and explanation
Answer: D. The slope is 5 dollars per ticket, so each additional ticket adds $5.
Linear equations in two variables
Equations and graphs of lines
Move between points, graphs, and equations.
Point-slope form
A line with slope m through (a, b) satisfies y − b = m(x − a). A point lies on a line only if substituting both coordinates makes the equation true.
Parallel and perpendicular
Distinct nonvertical parallel lines have equal slopes. Nonvertical perpendicular lines have slopes with product −1. Horizontal and vertical lines are a separate perpendicular pair.
A graph is a set of solutions
Every point (x, y) on a line satisfies its equation. A table samples some of those solutions; a graph shows their pattern. You can therefore test a proposed point by substitution without drawing the entire line, or use two distinct points to determine a nonvertical line’s slope and intercept.
Different forms expose different features
Slope-intercept form y = mx + b exposes rate and y-intercept. Point-slope form y − k = m(x − h) uses a known point. Standard form Ax + By = C can make intercepts easy to compute by setting x or y to zero. Rewriting changes the appearance, not the set of solutions.
Worked example
Find the line with slope 2 through (3, 7).
- Use y − 7 = 2(x − 3).
- Distribute and solve for y.
y = 2x + 1
Watch for this: A vertical line x = a has undefined slope; do not assign it a slope of zero.
Check your understanding
Which point lies on y = 3x − 2?
- (1, 3)
- (2, 4)
- (0, 2)
- (3, 6)
Show answer and explanation
Answer: B. At x = 2, y = 3(2) − 2 = 4, matching the point.
Systems of two linear equations in two variables
Solve and interpret linear systems
Find values that satisfy two equations at once.
Substitution and elimination
Substitute an expression from one equation into the other, or add scaled equations so a variable cancels. Then find the other coordinate.
Graph interpretation
One intersection means one solution. Distinct parallel lines have no solution. Equivalent equations describe the same line and have infinitely many solutions.
A solution must make both statements true
A system describes two constraints on the same unknowns. Each line contains infinitely many points, but only their common points satisfy the system. Substitution and elimination are ways to find those common values without testing every possible pair. An answer that works in just one equation is insufficient.
Choose a method from the structure
Substitution is efficient when a variable is already isolated. Elimination is efficient when coefficients are equal or opposite, or become so after a small multiplication. If you only need x + y or another combination, adding equations may reveal that quantity directly without solving separately for each variable.
Worked example
Solve x + y = 11 and x − y = 3.
- Add the equations to get 2x = 14.
- Then x = 7 and y = 11 − 7.
(x, y) = (7, 4)
Watch for this: An intersection is an ordered pair. Check whether the question wants x, y, or their sum.
Check your understanding
If x + y = 9 and x − y = 1, what is y?
- 1
- 5
- 4
- 8
Show answer and explanation
Answer: C. Adding gives 2x = 10, so x = 5 and y = 9 − 5 = 4.
Linear inequalities in one or two variables
Inequalities and feasible regions
Solve inequalities and interpret all allowed values.
Reverse when multiplying by a negative
Adding or subtracting preserves the inequality direction. Multiplying or dividing both sides by a negative reverses it.
Graph the allowed values
Strict inequalities use an open endpoint or dashed boundary; inclusive inequalities use a closed endpoint or solid boundary. A system requires the overlap of all conditions.
An inequality describes an ordered relationship
Instead of one equality, an inequality describes which side is greater or smaller. Its solution can be an interval or a region. “At most” includes the boundary and means ≤; “at least” means ≥. “More than” and “less than” are strict and exclude the boundary.
Why a negative factor reverses the sign
Multiplying by a negative reflects numbers across zero, reversing their order on a number line. Since 2 < 5, multiplying both values by −1 gives −2 > −5. The same reversal must occur when multiplying or dividing both sides of an inequality by a negative quantity.
Worked example
Solve −2x + 3 ≥ 11.
- Subtract 3: −2x ≥ 8.
- Divide by −2 and reverse the sign.
x ≤ −4
Watch for this: A shaded region contains many solutions. Test both inequalities before accepting a point.
Check your understanding
Which inequality is equivalent to −3x < 12?
- x < −4
- x > −4
- x > 4
- x < 4
Show answer and explanation
Answer: B. Divide both sides by −3 and reverse < to >, giving x > −4.
Put the ideas into practice
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