Subtopics

Linear equations in one variable

Solving equations of the form ax + b = c, including equations with fractions and variables on both sides.

Linear equations in two variables

Slope, intercepts, slope-intercept form, point-slope form, standard form, and contextual interpretation.

Systems of two linear equations

Substitution and elimination methods. Recognizing no-solution and infinite-solution cases.

Linear inequalities

Solving and graphing inequalities in one and two variables. Systems of linear inequalities.

Interpreting linear functions

Reading slope and intercept from context, identifying linear vs. non-linear models from tables and graphs.

Key formulas

y = mx + b (slope-intercept form)

y − y₁ = m(x − x₁) (point-slope form)

m = (y₂ − y₁) / (x₂ − x₁) (slope)

Ax + By = C (standard form)

New = Original × (1 ± r) (percent change)

See full formula sheet →

Test-taking tips

Always check: does the question want the value of x or an expression like 2x + 1?
When given a word problem, define your variable before writing the equation.
For systems, elimination is often faster than substitution when coefficients align.
On the SAT, "no solution" means parallel lines; "infinite solutions" means the same line.

Fully worked Algebra example

Worked example: Finding a base fee from a linear cost model

A reservation costs $198 for 11 hours. The hourly rate is $14 per hour, and the total cost includes a one-time base fee. What is the base fee?

Let b represent the one-time base fee. The total cost is the base fee plus the hourly charge:

b + 14(11) = 198

Calculate the charge for 11 hours:

14(11) = 154

Substitute that amount into the equation:

b + 154 = 198

Subtract 154 from both sides:

b = 198 - 154 = 44

The one-time base fee is $44.

Verification method: Substitute the base fee back into the cost model. 44 + 14(11) = 44 + 154 = 198, which matches the stated total.

Trap 1: Treating the total cost as an hourly amount. The $198 is the complete charge, not the amount charged each hour. Multiplying 198 by 11 ignores the structure of the situation.

Trap 2: Subtracting the hourly rate only once. The variable charge is $14 for each of 11 hours, so the amount to remove from the total is 14(11) = 154, not 14.

Why students get Algebra questions wrong

Losing track of what the variable represents.

Define the variable before building the equation, then keep that meaning fixed. In the reservation problem, b is the one-time base fee, not the total cost, the hourly rate, or the number of hours. After solving, answer with the quantity the question requested rather than a related intermediate value.

Dropping a sign while distributing.

A factor outside parentheses multiplies every term inside. For example, -3(x - 4) = -3x + 12 because (-3)(-4) is positive 12. Writing -3x - 12 changes the equation and usually produces a plausible-looking but incorrect result.

Forgetting when an inequality sign must flip.

Adding or subtracting the same value on both sides does not reverse an inequality. Multiplying or dividing both sides by a negative number does. For example, -2x > 8 becomes x < -4 after division by -2.

Confusing equal slopes with proportional equations in a system.

Equal slopes tell you that two lines are parallel or identical, but the intercepts decide which. In standard form, proportional coefficients for x and y alone indicate equal slopes. If the constant terms are not in the same proportion, the lines are distinct and the system has no solution. If all coefficients, including the constants, share the same proportion, both equations describe the same line and the system has infinitely many solutions.

Final Algebra check

Before accepting an answer, state what the variable represents, substitute the result into the original equation, and confirm that both sides match. For inequalities, also ask whether you multiplied or divided by a negative number. For systems, compare both the slopes and the constants before deciding that lines are parallel or identical.

Algebra.