SATStudy Topic
SAT Algebra Foundations Study Guide
Practice linear equations, inequalities, systems, ratios, and interpretations of slope and intercept.
Topic Overview
Identify what each term represents
A linear model combines a constant starting amount with a constant rate of change. In C = 18m + 24, the 24 is a one-time cost and 18 is the cost for each unit of m. Write the units beside the terms: dollars per month multiplied by months gives dollars. The intercept is the output when the input is zero, while slope compares a change in output with a change in input.
Solve and check
Use inverse operations consistently on both sides of an equation. For systems, choose substitution when one variable is already isolated and elimination when coefficients align. A contradiction such as 8 = 3 means no solution; an identity such as 8 = 8 after all variable terms cancel indicates infinitely many solutions. Reverse an inequality when multiplying or dividing by a negative number.
Worked example
A service costs a fixed $20 plus $6 per visit. With a budget of $68, solve 20 + 6v <= 68. Subtract 20 and divide by 6 to get v <= 8. Since visits are whole counts, eight is the largest allowed count. Check: 20 + 6(8) = 68. For percentage changes, similarly identify the starting base before dividing an increase or decrease by it.
- Using the new amount instead of the original amount as the percent-change base.
- Confusing slope with the y-intercept.
- Failing to reverse an inequality after division by a negative number.
- Reporting an algebraic answer without checking the original conditions.
Checkpoint Quiz
Test your understanding of SAT Algebra Foundations
These questions are for study practice only and are not official exam questions.
1. Which equation describes a line perpendicular to y = 2x + 1?
2. Solve 2(x - 3) + 4 = 3x - 7.
3. For the system y = 3x + 2 and 2y = kx + 9, which k makes the lines parallel with no common point?
4. A researcher models temperature by T = 0.8d + 15. What d corresponds to T = 31?
5. What is the x-intercept of the line 3x + 2y = 18?
6. Which inequality describes spending at most $150 on x notebooks costing $6 each and one bag costing $30?
7. A gym charges a $24 joining fee and $18 per month. Which expression gives the total cost in dollars for m months, including the joining fee?
8. What is the solution set of -2x + 5 > 13?
9. The table pairs x = 0, 1, 2, 3 with y = 7, 12, 17, 22. Which equation matches all four pairs?
10. If y varies directly with x and y = 18 when x = 6, what is y when x = 10?
Frequently asked questions
How do I find slope from two points?
Divide the change in y by the corresponding change in x, using the same point order for both differences.
What does a zero-input value mean?
It is the intercept of the model, often a starting amount or fixed charge.
When does a linear system have no solution?
For two nonvertical lines, equal slopes with different intercepts produce distinct parallel lines and no common solution.
How can I check a word-problem equation?
Substitute a simple input, verify the units, and see whether the model produces the stated starting amount and rate.
