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Math: Algebra

The PSAT's biggest math domain: 13 to 15 questions on linear equations, functions, systems and inequalities, every one of them drawn before it is solved.

40 topics · 160 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Fluently Solving a Linear Equation in One Variable

  1. How it worksHow does a linear equation collapse down to x equals a number, one legal move at a time?
  2. Worked exampleWalk me through solving 5(x - 3) = 2x + 9 for x in under 40 seconds.
  3. Worked exampleFind x when (2x + 7)/3 = 5 by clearing the fraction before anything else.
  4. Common mistakeWhy doesn't distributing the minus in 4 - 2(x - 5) give you 4 - 2x - 10?
  5. Going furtherWhat happens to the solution of 3x + 6 = 12 when every coefficient in it is doubled?
Your turneasy1 of 3
Solve (7x + 13)/7 = 6 for x. Enter your answer as a decimal if it is not a whole number.

Using Structure Instead of Grinding

  1. How it worksHow does spotting a repeated chunk let you treat it as a single variable?
  2. Common mistakeWhy doesn't multiplying out every bracket always get you to the answer faster?
  3. Worked exampleSolve 7(2x - 1) = 21 by dividing through by 7 before you expand anything.
  4. Worked exampleFind x when 6(x + 2) = 9(x + 2), treating x + 2 as one single block.
Your turneasy1 of 3
Solve 7(5x - 8) = 40 for x by dividing through by 7 before expanding. Enter your answer as a decimal if it is not a whole number.

Building a One-Variable Equation from a Story

  1. How it worksHow do a starting fee, a rate and a total turn into one equation in one variable?
  2. Worked exampleBuild and solve an equation for a gym costing 25 dollars to join plus 12 a month, billed 121.
  3. Worked exampleFind the number that becomes three times itself when you add 8 to it.
  4. Common mistakeWhy isn't the phrase 8 less than a number written as 8 - x?
Your turneasy1 of 3
A lab is billed a 48-dollar standing charge plus 7 dollars for every kilowatt-hour of electricity used. One month's bill is 383 dollars. Find the number of kilowatt-hours used.

Using the Equation to Answer What Was Actually Asked

  1. How it worksHow does writing down what x stands for stop you from answering a different question?
  2. Worked exampleFind how many texts make a 47-dollar bill when a plan costs 30 dollars plus 0.10 per text.
  3. Worked exampleFind the older sibling's age when two siblings are 6 years apart and their ages add to 38.
Your turneasy1 of 3
Two tanks hold 96 litres between them, and the first tank holds 21 litres more than the second. Find the volume in the SECOND tank.

What a Constant, Factor, or Term Stands For

  1. How it worksHow does each piece of C = 45 + 8h attach to something real in the story it came from?
  2. Worked exampleExplain what 0.9 and 220 mean in 220 - 0.9m = 40 for a tank draining over m minutes.
  3. Common mistakeWhy isn't the largest number in an equation automatically the starting amount?
  4. Going furtherWhat changes about a coefficient's meaning when the units switch from hours to minutes?
Your turneasy1 of 3
A pool's depth in centimetres m minutes after the drain opens is D = 254 - 7m. What does 7 represent?

No Solution, One Solution, or Infinitely Many

  1. How it worksHow does the x term vanishing tell you instantly which of the three cases you landed in?
  2. Worked exampleFind the value of c that makes 4x + c = 4x + 11 true for every single x.
  3. Worked exampleShow me line by line why 6x + 5 = 6x - 2 can never be satisfied.
  4. Common mistakeWhy doesn't a solution of x = 0 count as having no solution?
  5. Going furtherWhat has to be true about a and b for ax + 3 = bx + 3 to have exactly one solution?
Your turneasy1 of 3
Find the value of k for which 7(x + 11) = 7x + k is true for every value of x.

Evaluating a Linear Function and Running It Backwards

  1. How it worksHow does f(x) = 3x - 7 process a number, operation by operation, into its output?
  2. Worked exampleFind f(5) for f(x) = 3x - 7, then find the input that makes the output 20.
  3. Common mistakeWhy doesn't f(2x) mean the same thing as twice f(x) for a line with a nonzero intercept?
  4. Worked exampleFind g(-4) and the x value making g(x) equal 1, where g(x) = -2x + 11.
Your turneasy1 of 3
For f(x) = 7x - 14, find the input x that makes f(x) = 57. Enter your answer as a decimal if it is not a whole number.

Writing a Linear Rule from Two Input-Output Pairs

  1. How it worksHow does the rise over the run between two pairs become the number in front of x?
  2. Worked exampleBuild the rule for a linear function with f(1) = 5 and f(4) = 17.
  3. Worked exampleBuild the rule for a linear function that starts at 9 and drops 4 for every step of 1 in x.
  4. Going furtherWhat happens to the rule when the two input-output pairs share the very same output?
Your turneasy1 of 3
A linear function f satisfies f(0) = 15 and f(6) = 77. Write the rule for f(x).

Modelling a Relationship with a Linear Function

  1. How it worksHow do you decide which quantity in a story is the input and which one is the output?
  2. Worked exampleBuild a function for a candle 30 cm tall that burns 2.5 cm every hour.
  3. Common mistakeWhy isn't every steadily growing quantity described by a linear function?
Your turneasy1 of 3
A rain barrel already holds 45 litres and collects 7 litres for each hour of steady rain. Write an expression in terms of h for the litres in the barrel after h hours of rain.

Using a Linear Function to Answer a Context Question

  1. How it worksHow does feeding a future input into a model turn plain algebra into a forecast?
  2. Worked exampleFind when a pool holding 400 gallons and draining 15 a minute is down to 100 gallons.
  3. Worked exampleFind the fare for a 14-mile ride when a taxi charges 3.50 dollars plus 2.25 per mile.
  4. Going furtherWhat happens to a model's prediction once the input leaves the range the data covered?
Your turneasy1 of 3
A ride-share trip costs 6 dollars to start plus 4 dollars for each mile driven. Find the cost of a 19-mile trip.

Interpreting a Slope, a Starting Value, or an Input-Output Pair

  1. How it worksHow does f(0) always land on the vertical axis and become the starting amount?
  2. Worked exampleExplain what P(0) and the 45 say about a town's population when P(t) = 1200 + 45t.
  3. Common mistakeWhy isn't a negative slope the same thing as a negative output value?
  4. Going furtherWhat changes about a slope's meaning when the model runs backwards in time?
Your turneasy1 of 3
A town's population t years after 2020 is modelled by P(t) = 2362 + 59t. What does 59 tell you?

Reading a Linear Function's Graph in Context

  1. How it worksHow do a line's steepness and its vertical crossing become a rate and a starting value?
  2. Worked exampleFind the hourly rate of a line starting at 60 on the vertical axis and hitting 0 at hour 12.
  3. Worked exampleFind the output at x = 7 for the line through (2, 5) and (10, 29).
  4. Common mistakeWhy doesn't a steeper-looking line always mean a bigger rate when the axis scales differ?
Your turneasy1 of 3
On a graph of fuel against time, the line crosses the vertical axis at 96 litres and reaches zero litres at hour 11. Find the number of litres used per hour.

Table, Rule, and Graph of a Linear Function

  1. How it worksHow do you test a table for a constant rate before assuming it came from a line?
  2. Worked exampleFind the rule behind a table where x of 0, 1, 2, 3 gives y of -5, -1, 3, 7.
  3. Worked exampleFind which of x + 2y = 10 or y = 2x + 10 matches a line through (0, 5) and (10, 0).
  4. Common mistakeWhy doesn't a table with equal y-steps have to be linear when the x-steps are uneven?
  5. Going furtherWhat does a table look like when the line behind it runs perfectly horizontal?
Your turneasy1 of 3
A table shows that y equals -8 when x is 0, and that y rises by 7 for every step of 1 in x. Write the rule for y in terms of x.

Matching a Table, a Rule, and a Graph in a Story

  1. How it worksHow do the units of a context survive the trip from a table across to a graph?
  2. Worked exampleBuild the rule and mark its intercept when 2 hours costs 44 dollars and 5 hours costs 95.
  3. Going furtherWhat happens to a context graph when only whole-number inputs make any sense?
Your turnmedium1 of 3
A repair shop's charge is linear in the hours booked: 3 hours costs 49 dollars and 10 hours costs 110 dollars. Write an expression in terms of h for the charge on an h-hour booking.

From an Input to an Output, and Back Again

  1. How it worksHow does reversing a linear rule undo each operation in the opposite order?
  2. Worked exampleFind m when the charge C = 18 + 0.25m comes to 46.75 dollars.
  3. Worked exampleFind the t where h(t) = 64 - 16t reaches 0, and say what that moment means.
  4. Common mistakeWhy isn't every output value reachable when a story restricts which inputs are allowed?
Your turneasy1 of 3
A storage locker costs C = 31 + 9m dollars after m months. Find m when the charge C reaches 286 dollars. Enter your answer as a decimal if it is not a whole number.

Creating and Using a Linear Equation in Two Variables

  1. How it worksHow does a budget split between two items become a single equation in x and y?
  2. Common mistakeWhy isn't one of the two variables in Ax + By = C automatically the input?
  3. Worked exampleBuild an equation for spending exactly 60 dollars on 4-dollar notebooks and 6-dollar binders.
  4. Worked exampleFind how many binders 4n + 6b = 60 allows once 3 notebooks are bought.
Your turneasy1 of 3
A student spends exactly 145 dollars on n notebooks costing 5 dollars each and k binders costing 10 dollars each. Which equation relates n and k?

Modelling a Constraint on Two Quantities

  1. How it worksHow does each one costs this much and together they make that become Ax + By = C?
  2. Worked exampleBuild a constraint for seats priced at 8 and 12 dollars together raising exactly 2400.
  3. Going furtherWhat changes in the equation when the two quantities are measured in different units?
Your turnhard1 of 3
A van is loaded with x boxes of 16 kilograms each and y crates of 11 POUNDS each, filling its 866-kilogram limit exactly. One pound is 0.4536 kilograms. Write an expression in terms of x for y.

Interpreting a Solution, Constant, or Term in Context

  1. How it worksHow does the constant on the right of Ax + By = C carry the total from the story?
  2. Worked exampleExplain what 5, 3 and 180 mean in 5s + 3l = 180 for boxes totalling 180 kilograms.
  3. Common mistakeWhy isn't the coefficient of x in a constraint equation a rate of change?
  4. Going furtherWhat happens to a solution point's meaning when one coordinate must be a whole number?
Your turneasy1 of 3
A pallet is packed with s small boxes and l large boxes, and 7s + 17l = 385 records its total mass in kilograms. What does 17 stand for?

Reading the Graph of Ax + By = C in Context

  1. How it worksHow does setting one variable to zero at a time hand you both intercepts of Ax + By = C?
  2. Worked exampleFind both intercepts of 4x + 10y = 200 and say what each one means for a 200-dollar budget.
  3. Common mistakeWhy isn't the C in Ax + By = C the same as the line's vertical intercept?
  4. Worked exampleFind the slope of 6x + 3y = 18 by rewriting it with y alone on the left.
Your turneasy1 of 3
Find the y value at which the line 7x + 9y = 192 crosses the vertical axis.

From the Equation of a Line to Its Graph and Back

  1. How it worksHow do you plot a line from its equation using only the intercept and a single step?
  2. Worked exampleGraph y = -3x + 4 by marking the intercept and stepping down 3 across 1.
  3. Worked exampleBuild the equation of a line crossing the vertical axis at -2 and rising 5 for every 2 across.
  4. Common mistakeWhy doesn't a bigger value of b make a line steeper?
Your turneasy1 of 3
A line crosses the vertical axis at -9 and rises 11 units for every 6 units across. Write its equation, giving y in terms of x.

From a Table to the Equation of a Line

  1. How it worksHow do you pull a slope out of a table whose x values skip around unevenly?
  2. Worked exampleFind the equation of the line passing through the pairs (-2, 13) and (3, -2).
  3. Common mistakeWhy doesn't the top row of a table have to be the line's vertical intercept?
Your turneasy1 of 3
A table's two rows read x = -4, y = 20 and x = 8, y = -16. Write the equation of the line, giving y in terms of x.

Table, Equation, and Graph of a Constraint in Context

  1. How it worksHow do you decide which stretch of a constraint line actually makes sense in a story?
  2. Worked exampleShow me every whole-number pair on 2c + 5a = 40 and where each one sits on the line.
  3. Going furtherWhat happens to the picture when the story allows fractional amounts of a quantity?
  4. Ties togetherHow does a single constraint line turn into a system the moment a second condition appears?
Your turneasy1 of 3
A blend uses x kilograms of oats at 4 dollars a kilogram and y kilograms of nuts at 14 dollars a kilogram, costing exactly 192 dollars. Find the kilograms of oats in the blend that uses no nuts.

Given One Quantity, Finding the Other

  1. How it worksHow does substituting a known value collapse a two-variable equation into a one-variable one?
  2. Worked exampleFind the value of y in 7x + 4y = 96 when x is 8.
  3. Worked exampleFind q in 3p + 5q = 70 when p is 10, and say whether that count could be a whole one.
Your turneasy1 of 3
Find the value of y in 8x + 6y = 212 when x is 6.

Writing a Line's Equation from Two Points or a Point and a Slope

  1. How it worksHow does point-slope form let you write a line without hunting for b first?
  2. Worked exampleFind the equation of the line passing through (2, -1) and (6, 7).
  3. Worked exampleFind the equation of the line through (-3, 5) whose slope is -2.
  4. Common mistakeWhy isn't slope the change in x divided by the change in y?
Your turneasy1 of 3
Write the equation of the line through (7, -8) with slope -7, giving y in terms of x.

Writing a Line Parallel or Perpendicular to Another

  1. How it worksHow does rotating a line by 90 degrees flip its slope upside down and change the sign?
  2. Worked exampleFind the line through (4, 1) that runs parallel to y = -3x + 8.
  3. Worked exampleFind the line through (6, -2) perpendicular to y = (2/3)x - 5.
  4. Common mistakeWhy isn't a perpendicular slope simply the negative of the original slope?
  5. Going furtherWhat happens to the perpendicular slope when the original line is perfectly horizontal?
Your turneasy1 of 3
Write the equation of the line through (7, 8) that runs parallel to y = -7x + 6, giving y in terms of x.

Solving a System by Substitution and by Elimination

  1. How it worksHow does elimination make a whole variable disappear by adding two equations together?
  2. How it worksHow does substitution trade one equation's variable for an expression taken from the other?
  3. Worked exampleSolve 3x + 2y = 16 together with x - 2y = 0 by adding the two equations.
  4. Worked exampleSolve y = 2x - 1 together with 5x + 3y = 30 by substitution.
  5. Common mistakeWhy doesn't finding x finish the job when the question asked for x plus y?
Your turneasy1 of 3
Solve the system 7x + 6y = 57 and x - 6y = 5 by adding the two equations. Find x.

Structure Shortcuts in a System

  1. How it worksHow does adding or subtracting two equations produce exactly the combination you were asked for?
  2. Worked exampleFind x + y when 4x + 3y = 22 and 3x + 4y = 20, without ever isolating x.
  3. Worked exampleFind the value of 2a - 2b when 5a + b = 17 and 3a + 3b = 15.
  4. Going furtherWhat changes when the two equations are scaled by different numbers before combining?
Your turneasy1 of 3
Given 10x + 4y = 57 and 4x + 10y = 25, find the value of x + y.

Creating and Using a System in Context

  1. How it worksHow does each sentence of a two-unknown story become one row of a system?
  2. Worked exampleFind each ticket count when 90 tickets priced at 12 and 7 dollars raise 855 dollars.
  3. Worked exampleFind the two numbers that add up to 41 and differ by 9.
  4. Common mistakeWhy isn't a single equation enough when a story names two different quantities?
Your turneasy1 of 3
Two numbers add up to 96 and differ by 21. Find the larger number.

Modelling Two Constraints with a System

  1. How it worksHow do a count constraint and a value constraint end up as two different lines?
  2. Worked exampleBuild a system for 18 coins of nickels and dimes together worth 1.45 dollars.
  3. Going furtherWhat happens to the picture when a third constraint is piled onto the same two unknowns?
Your turnmedium1 of 3
A jar holds 25 coins, each worth either 4 cents or 24 cents, and the coins are worth V cents altogether. Write an expression in terms of V for the number of 4-cent coins.

When a System Has No Solution or Infinitely Many

  1. How it worksHow does matching slopes with different intercepts force two lines to stay apart forever?
  2. How it worksHow does one equation being a multiple of the other land the two lines exactly on top of each other?
  3. Worked exampleFind the k that gives 2x + ky = 9 and 6x + 12y = 27 infinitely many solutions.
  4. Worked exampleShow me why 5x - y = 3 and 10x - 2y = 14 cannot both be satisfied.
  5. Common mistakeWhy doesn't a system with no solution mean you slipped up somewhere in the algebra?
Your turneasy1 of 3
The equation 7x + 8y = 38 is multiplied through by 3 to produce a second equation describing the very same line. Find the constant on the right of that second equation.

A System and Its Graph

  1. How it worksHow does testing a candidate point in both equations confirm it really is the solution?
  2. Worked exampleFind where y = x + 2 meets y = -2x + 11, then check the point in both equations.
  3. Common mistakeWhy isn't a point that satisfies only one of the equations a solution of the system?
  4. Worked exampleFind how many solutions 3x - y = 4 and -6x + 2y = -8 have by graphing them.
Your turneasy1 of 3
Find the x-coordinate of the point where y = x + 7 meets y = -5x + 57.

A System and Its Graph in a Story

  1. How it worksHow do you read which plan is cheaper from which side of the crossing you sit on?
  2. Worked exampleFind where a plan costing 40 plus 5 a class ties with one costing 25 plus 8 a class.
  3. Going furtherWhat does it mean when two cost lines in a story never cross for any positive input?
  4. Ties togetherHow does the tie point between two cost lines connect to solving an inequality between them?
Your turneasy1 of 3
Plan A costs 54 dollars plus 6 dollars a class and Plan B costs 29 dollars plus 11 dollars a class. Find the number of classes at which the two plans cost the same.

Solving a Linear Inequality in One Variable

  1. How it worksHow does dividing by a negative number flip which way the inequality sign points?
  2. How it worksHow does a number line turn an inequality's answer into a ray with an open or a closed end?
  3. Worked exampleSolve -4x + 9 > 25 and draw the answer on a number line.
  4. Worked exampleSolve 3(x - 2) ≤ 5x + 4 and name the smallest integer that works.
  5. Common mistakeWhy doesn't subtracting a negative number flip the inequality sign?
Your turneasy1 of 3
Solve -7x + 15 > 58. Find the boundary value of x at which the solution set stops.

Creating and Using Linear Inequalities in Context

  1. How it worksHow does a spending ceiling become an inequality rather than an equation?
  2. Worked exampleFind the longest whole mileage 30 dollars buys when a ride costs 3.50 plus 1.75 a mile.
  3. Worked exampleFind the fewest 6-dollar shifts needed to earn at least 250 dollars.
  4. Common mistakeWhy isn't more than 10 the same as at least 10 once the answer must be whole?
Your turneasy1 of 3
A ride costs 6 dollars plus 4 dollars a mile, and a rider has 87 dollars. Find the greatest mileage the budget covers, as a decimal if it is not whole.

Modelling a Constraint with an Inequality in Two Variables

  1. How it worksHow does a cap on two purchases turn into Ax + By no greater than C?
  2. Worked exampleBuild an inequality for buying x items at 4 dollars and y at 9 on a 72-dollar budget.
  3. Going furtherWhat changes about the region when the budget has to be spent exactly?
Your turneasy1 of 3
A shopper buys x items at 6 dollars each and y items at 12 dollars each, spending at most 135 dollars. Which inequality models that?

Which Half-Plane to Shade

  1. How it worksHow does dropping a single test point decide the shading for an entire half-plane?
  2. Worked exampleShow me the solution region of y > 2x - 3 using the origin as the test point.
  3. Worked exampleFind whether (4, 1) satisfies 3x - 5y ≤ 8, then shade the matching side.
  4. Common mistakeWhy isn't the region above the boundary always right when the sign says greater than?
  5. Going furtherWhat changes about the boundary line itself when the inequality becomes strict?
Your turneasy1 of 3
The inequality y > 5x - 9 is graphed and the origin is used as the test point. What happens?

Testing a Point Against a Solution Set

  1. How it worksHow does checking a point against two inequalities at once locate the overlap region?
  2. Worked exampleFind whether (2, 5) satisfies both y ≥ x + 1 and 2x + y ≤ 10.
  3. Worked exampleFind an integer point sitting inside the overlap of y < -x + 6 and y > x - 2.
  4. Common mistakeWhy isn't a point sitting on a dashed boundary part of the solution set?
Your turneasy1 of 3
To test (7, 8) against 7x + y ≤ 31, find the value of 7x + y at that point.

Table, Algebra, and Graph of an Inequality

  1. How it worksHow do you recover an inequality from the edge between its true and false points?
  2. Worked exampleBuild the inequality with boundary y = 2x + 1 whose region holds (0, 5) but not (3, 2).
  3. Worked exampleFind three ordered pairs that satisfy 2x + y < 8 and one pair that fails it.
  4. Going furtherWhat happens to the shaded region when the boundary line runs straight up and down?
Your turnmedium1 of 3
An inequality has boundary y = 5x + 7. The point (0, 39) satisfies it and the point (4, -19) fails it. Which inequality is it?

Student-Produced Responses: Entering the Answer

  1. How it worksHow does a six-character entry box force a choice between a fraction and a decimal?
  2. How it worksHow does the answer preview in the testing app catch an entry that is not what you meant?
  3. Worked exampleShow me every legal way to enter an answer of seven halves into the response box.
  4. Common mistakeWhy doesn't typing 3 1/2 mean three and a half, and what does the scorer read instead?
  5. Common mistakeWhy isn't a dollar sign, a comma or a percent sign ever allowed in the answer box?
Your turneasy1 of 3
A student's answer is 7 and one half, and they type 7 1/2 into the response box. How is that entry read?

When a Student-Produced Response Has More Than One Right Answer

  1. How it worksHow does a range-style question end up with a whole stretch of acceptable entries?
  2. Worked exampleFind one value of x satisfying 12 < 4x < 30 that would be scored correct.
  3. Common mistakeWhy doesn't entering two answers separated by a comma earn the point?
Your turneasy1 of 3
A question accepts any x satisfying 17 < 5x < 77. Find the midpoint of the range of acceptable values of x.

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