The companion set to Workshop 7. Twelve real SAT-style questions on linear equations and algebra. Don't grind, work the shortcut: interpret the model, substitute, or graph it.
SAT algebra rewards doing less math, smarter. Before you solve, pick the move that fits the question: interpret what each number means in slope-intercept or standard form; substitute by testing the answer choices or picking your own numbers; or graph it and let Desmos find the point or the number of solutions. Here is how the wrong answers try to trip you:
Swapped Role. Treats the slope (the rate) as the intercept (the starting value), or the reverse.
Evaluated Instead. Plugs in a value and reports an output when the question asks what a number means.
Translation Slip. Builds the equation with the flat fee as a rate, or drops a term entirely.
Direction Flipped. Reverses an inequality sign, or the direction of an increase or decrease.
Only Half the System. A point or model that satisfies one equation or inequality but not both.
Coefficient Swap. Reads the wrong coefficient, swapping the slope and intercept or two numbers in a form.
Question 1 of 12
Interpret a ModelMedium
A print shop uses the equation
$c = 0.07p + 3.50$
to determine the cost $c$, in dollars, of an order in which $p$ pages are printed.
Which of the following is the best interpretation of $3.50$ in this context?
The takeaway: The constant term is the model's value at $p = 0$: a charge of 3.50 dollars that applies to an order regardless of the number of pages. The coefficient of $p$ shows each printed page then adds 0.07 dollars.
Build the EquationMedium
A bike-share plan charges 6 dollars to unlock a bike for the day plus 3 dollars for each mile ridden. Let $C(m)$ be the total cost, in dollars, of riding $m$ miles in a day.
Which function represents this situation?
The takeaway: The 3-dollar per-mile rate multiplies $m$, and the 6-dollar unlock fee is added once. So $C(m) = 3m + 6$. At $m = 7$ this gives $27$.
Systems & InequalitiesMedium
A school is arranging vans and cars to transport students to a competition. Each van seats 8 students and each car seats 4 students. The school must transport at least 60 students and can use fewer than 12 vehicles.
Which of the following systems of inequalities represents this situation, where $v$ is the number of vans and $c$ is the number of cars used?
The takeaway: Seating is $8v + 4c$ students, since each of the $v$ vans seats 8 and each of the $c$ cars seats 4; "at least 60" gives $8v + 4c \ge 60$. The vehicle count is $v + c$, and "fewer than 12" excludes 12 itself, giving the strict $v + c < 12$.
Interpret a ModelMedium
Ms. Rivera spends 25 minutes grading each quiz and 40 minutes grading each lab report. The equation below relates $q$, the number of quizzes she graded, and $r$, the number of lab reports she graded, one weekend.
$25q + 40r = 400$
What is the best interpretation of the number $400$ in this equation?
The takeaway: In the equation $25q + 40r = 400$, the term $25q$ is the time spent grading quizzes and $40r$ is the time spent grading lab reports. The equation states that these two amounts of time add up to 400, so 400 is the total number of minutes spent grading that weekend.
Build the EquationMedium
A photo lab charges a flat setup fee of 8 dollars per order plus 2 dollars for each print. Let $c$ be the total cost, in dollars, of an order containing $p$ prints.
Which equation represents this situation?
The takeaway: The 2-dollar per-print rate multiplies $p$, and the 8-dollar setup fee is added once. So $c = 2p + 8$. At $p = 10$ this gives $28$.
Systems & InequalitiesMedium
Mara knits scarves and blankets to sell. Each scarf uses 2 yards of yarn and takes 1 hour to make, and each blanket uses 6 yards of yarn and takes 4 hours to make. Mara has at most 48 yards of yarn and at most 20 hours available. Let $x$ be the number of scarves and $y$ be the number of blankets she makes.
Which of the following systems of inequalities represents this situation?
The takeaway: The yarn used is 2 yards per scarf plus 6 yards per blanket, so having at most 48 yards means $2x + 6y \le 48$. The time used is 1 hour per scarf plus 4 hours per blanket, so having at most 20 hours means $x + 4y \le 20$.
Interpret a ModelChallenge
A botanist tracks a bamboo shoot that grows at a constant rate. The height $H(d)$, in centimeters, of the shoot $d$ days after the start of the study is given by
$H(d) = 0.8d + 30$
Which statement best compares the meanings of 0.8 and 30 in this context?
The takeaway: At the start of the study, $d = 0$ and $H(0) = 30$, so the shoot begins at 30 centimeters. The coefficient 0.8 multiplies the days, so the shoot adds 0.8 centimeters of height each day.
Build the EquationChallenge
A phone plan costs 32 dollars per month plus 6 cents for each minute of international calling.
Which equation gives the number of international minutes $m$ used in a month in terms of that month's total cost $C$, in dollars?
The takeaway: Since $C$ is in dollars, the 6-cent rate must be written as $0.06$ dollars per minute: $C = 32 + 0.06m$. Solving for $m$ gives $m = \dfrac{C - 32}{0.06}$. Leaving the rate as 6 while working in dollars mixes units and shrinks the answer by a factor of 100.
Systems & InequalitiesChallenge
In the xy-plane, the point $(-2, 5)$ is a solution to which of the following systems of inequalities?
The takeaway: Substitute $x = -2$ and $y = 5$ into each system; the point must make both inequalities of a system true. For the correct system, $5 > -2 + 4 = 2$ is true and $5 \le -3(-2) + 1 = 7$ is true.
Interpret a ModelChallenge
Keiko has a dog-walking job that pays a flat weekly amount for supplies plus a set amount for each walk. The equation below models her total earnings $E$, in dollars, in a week during which she completes $n$ walks.
$E = 18n + 25$
What is the best interpretation of $18$ in this context?
The takeaway: The coefficient of $n$ is the amount earned per walk: each walk adds 18 dollars to Keiko's weekly earnings. The 25 is the flat weekly supply payment she receives even in a week with no walks.
Build the EquationChallenge
At a farm stand, a crate of peaches costs 4 dollars less than twice the cost of a crate of plums. A restaurant bought one crate of each for a total of $d$ dollars.
Which equation gives the cost $p$, in dollars, of a crate of plums in terms of $d$?
The takeaway: A crate of peaches costs $2p - 4$ dollars, so the pair costs $p + (2p - 4) = d$, which gives $3p - 4 = d$. Solving for $p$ yields $p = \dfrac{d + 4}{3}$. The subtraction direction (4 less than twice) and remembering to move the $-4$ across as $+4$ are where this item punishes a rushed setup.
Systems & InequalitiesChallenge
$y < 3x - 2$
$y \ge x + 1$
Which of the following points is NOT a solution to the given system of inequalities in the xy-plane?
The takeaway: Test each point in both inequalities. The point $(2, 4)$ gives $4 < 3(2) - 2 = 4$, which is false because the first inequality is strict, so $(2, 4)$ lies on an excluded boundary line and is not a solution. Each of the other points satisfies both inequalities; in particular $(3, 4)$ lies on the line $y = x + 1$, whose boundary is included by $\ge$.
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