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PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Algebra Syllabus

The Math: Algebra syllabus for PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) in full: 4 chapters, 12 topics and 15 sub-topics, every one listed below, plus 50 Math: Algebra flashcards with answers.

4Chapters
12Topics
15Sub-topics
50Flashcards

Math: Algebra syllabus: all 4 chapters

Expand any chapter to see its topics and sub-topics.

  1. Linear Equations in One Variable

     3 topics
    • Solving Single-Variable Linear Equations
      • Equations with variables on both sides
      • Equations with fractions and parentheses
    • Creating Linear Equations from Context
      • Translating word problems into equations
    • Equations with No or Infinite Solutions
      • Conditions for identity and contradiction

  2. Linear Equations in Two Variables

     3 topics
    • Slope-Intercept and Standard Form
      • Interpreting slope and intercepts in context
      • Converting between forms
    • Graphing Lines
      • Plotting from equations and tables
      • Finding equations from graphs
    • Writing Linear Models
      • Modeling real-world linear relationships

  3. Systems of Linear Equations

     3 topics
    • Solving by Substitution and Elimination
      • Choosing an efficient method
    • Number of Solutions of a System
      • Parallel, intersecting, and identical lines
    • Systems from Word Problems
      • Setting up two-equation models

  4. Linear Inequalities

     3 topics
    • Solving One- and Two-Variable Inequalities
      • Flipping the inequality sign
    • Graphing Inequalities and Systems
      • Identifying solution regions
    • Inequalities in Context
      • Constraints and feasible values

PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Algebra flashcards — 50 questions with answers

  1. What is a single-variable linear equation?

    An equation that can be written as ax + b = 0, where the variable appears only to the first power and a ≠ 0; it has exactly one solution.

  2. What are the basic steps to solve a single-variable linear equation?

    1) Distribute and clear parentheses, 2) combine like terms on each side, 3) move variable terms to one side and constants to the other using addition/subtraction, 4) divide by the coefficient of the variable.

  3. To solve a linear equation with fractions, what is the most efficient first step?

    Multiply both sides by the least common denominator (LCD) of all fractions to clear the denominators.

  4. Solve: 3(x − 4) = 2x + 5.

    3x − 12 = 2x + 5, so x = 17.

  5. When solving a literal equation for one variable in terms of others, what rule governs each step?

    Apply inverse operations to both sides (treat the target variable like the unknown), isolating it just as you would a number-only equation.

  6. When translating a word problem into a linear equation, what does the phrase 'per' typically indicate?

    A constant rate, which becomes the coefficient (slope) multiplying the variable.

  7. In a linear model, what does a fixed one-time amount (like a flat fee or starting value) become?

    The constant term (the y-intercept) of the equation.

  8. A gym charges a $25 sign-up fee plus $15 per month. Write an equation for total cost C after m months.

    C = 15m + 25.

  9. How do you find when two linear cost models are equal (a break-even point)?

    Set the two expressions equal to each other and solve for the variable.

  10. Translate: 'five less than twice a number is 11' into an equation.

    2x − 5 = 11.

  11. How can you tell a linear equation has NO solution?

    After simplifying, the variables cancel out and you get a false statement (e.g., 3 = 7).

  12. How can you tell a linear equation has INFINITELY many solutions?

    After simplifying, the variables cancel out and you get a true statement (e.g., 5 = 5), meaning both sides are identical (an identity).

  13. For ax + b = cx + d, what condition gives exactly one solution?

    a ≠ c (the coefficients of x differ).

  14. For ax + b = cx + d, what conditions give no solution?

    a = c but b ≠ d (same slope, different constant).

  15. For ax + b = cx + d, what conditions give infinitely many solutions?

    a = c and b = d (the two sides are identical).

  16. In the equation 6x + 12 = a(2x + 4), what value of a creates infinitely many solutions?

    a = 3, because then both sides equal 6x + 12.

  17. What is the slope-intercept form of a line?

    y = mx + b, where m is the slope and b is the y-intercept.

  18. What is the standard form of a linear equation?

    Ax + By = C, where A, B, and C are constants (commonly integers, A ≥ 0).

  19. How do you find the slope from standard form Ax + By = C?

    Slope = −A/B.

  20. How do you find the y-intercept from standard form Ax + By = C?

    y-intercept = C/B (set x = 0).

Show the other 30 Math: Algebra flashcards

  1. What is the point-slope form of a line?

    y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a point on the line.

  2. What is the slope formula given two points (x₁, y₁) and (x₂, y₂)?

    m = (y₂ − y₁) / (x₂ − x₁).

  3. What does the slope of a line represent in a real-world context?

    The constant rate of change—how much y changes for each one-unit increase in x.

  4. What is the slope of a horizontal line, and what is its equation form?

    Slope = 0; equation is y = constant.

  5. What is the slope of a vertical line, and what is its equation form?

    Slope is undefined; equation is x = constant.

  6. How are the slopes of two parallel lines related?

    They are equal (m₁ = m₂).

  7. How are the slopes of two perpendicular lines related?

    They are negative reciprocals; their product is −1 (m₁ · m₂ = −1).

  8. To graph a line from y = mx + b, what is the standard procedure?

    Plot the y-intercept (0, b), then use the slope m = rise/run to find a second point, and draw the line through them.

  9. How do you find the x-intercept of a line?

    Set y = 0 and solve for x.

  10. A line passes through (0, 4) with slope −2. Write its equation.

    y = −2x + 4.

  11. Write the equation of the line through points (1, 2) and (3, 8).

    Slope = (8−2)/(3−1) = 3; y = 3x − 1.

  12. What is a system of linear equations?

    Two or more linear equations considered together; a solution is an ordered pair (x, y) that satisfies all equations simultaneously.

  13. What does the solution to a system of two linear equations represent graphically?

    The point of intersection of the two lines.

  14. Describe the substitution method for solving a system.

    Solve one equation for one variable, substitute that expression into the other equation, solve for the remaining variable, then back-substitute to find the first.

  15. Describe the elimination method for solving a system.

    Multiply equations as needed so one variable has opposite coefficients, add the equations to eliminate that variable, solve, then back-substitute.

  16. When is substitution usually the easier method?

    When one variable is already isolated or has a coefficient of 1 (or −1).

  17. Solve by substitution: y = 2x and x + y = 9.

    x + 2x = 9, so x = 3 and y = 6; solution (3, 6).

  18. Solve by elimination: x + y = 10 and x − y = 4.

    Adding gives 2x = 14, so x = 7 and y = 3; solution (7, 3).

  19. How many solutions does a system have when the lines have different slopes?

    Exactly one solution (the lines intersect at one point).

  20. How many solutions does a system have when the lines have the same slope but different y-intercepts?

    No solution (the lines are parallel and never intersect).

  21. How many solutions does a system have when both equations represent the same line?

    Infinitely many solutions.

  22. While solving a system algebraically, you get 0 = 5. What does this mean?

    The system has no solution (the lines are parallel).

  23. While solving a system algebraically, you get 0 = 0. What does this mean?

    The system has infinitely many solutions (the equations represent the same line).

  24. For what value of k does the system y = 3x + 2 and y = kx − 5 have no solution?

    k = 3 (equal slopes with different intercepts make the lines parallel).

  25. Tickets cost $8 for adults and $5 for children. 200 tickets sold for $1,300. Write the system.

    a + c = 200 and 8a + 5c = 1300, where a = adult and c = child tickets.

  26. In a mixture or word-problem system, what two equations do you usually write?

    One equation for the total quantity (counts/amounts) and one for the total value (cost, weight, or concentration).

  27. What are the basic steps to solve a linear inequality in one variable?

    Isolate the variable using inverse operations, just like an equation—but flip the inequality sign whenever you multiply or divide both sides by a negative number.

  28. Solve: −2x + 3 ≤ 9.

    −2x ≤ 6, divide by −2 and flip: x ≥ −3.

  29. How do you graph the solution to a two-variable linear inequality?

    Graph the boundary line (solid for ≤ or ≥, dashed for < or >), then shade the side that satisfies the inequality (test a point such as (0,0)).

  30. What does the overlapping shaded region represent when graphing a system of inequalities, and how is 'at least'/'no more than' translated?

    The overlap is the set of points satisfying all inequalities; 'at least' means ≥, 'at most'/'no more than' means ≤, and quantities are typically also restricted to x ≥ 0, y ≥ 0.

PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Algebra FAQ

What is in the PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Algebra syllabus?

PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Algebra is split into 4 chapters: Linear Equations in One Variable, Linear Equations in Two Variables, Systems of Linear Equations and Linear Inequalities. Together they contain 12 topics and 15 sub-topics.

How many chapters are in PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Algebra?

4 chapters, containing 12 topics and 15 sub-topics. The largest chapter is Linear Equations in One Variable, with 3 topics.

Are there flashcards for PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Algebra?

Yes — a 50-card Math: Algebra deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.