SAT Slope-Intercept Word Problems: Units, Starting Values, and Negative Slopes

Interpret slope, units, and a starting value in an original poster-shop model, including a balance that decreases. Four practice questions have checked answers.

KiboNovember 13, 2026

In a linear model $y=mx+b$ , the slope $m$ is the change in $y$ for each increase of $1$ in $x$ , and $b$ is the value of $y$ when $x=0$ . Give $b$ the units of $y$ . Give $m$ the units of $y$ per unit of $x$ . A negative slope means $y$ decreases as $x$ increases. In a price, the slope is the amount added for one more item, and the intercept is the charge at an item count of zero when that count is part of the pricing rule.

College Board’s SAT Algebra question types include linear equations in two variables and linear functions. The Student Question Bank describes linear functions as models whose constants and rate of change can be interpreted, and linear equations in two variables as equations whose slope can be found and whose line can be written from given information. OpenStax Elementary Algebra explains slope-intercept form: the coefficient of the input is the slope, and the constant term is the vertical coordinate of the intercept. The shop below is original Kibo practice.

Read the poster price in slope-intercept form

A campus poster shop charges a $\$18$ design fee plus $\$0.75$ for every poster. For $n$ posters, the cost in dollars is

$C(n)=0.75n+18$

Here $0.75=\frac{3}{4}$ . The shop accepts every whole-number order, including a design-only request with zero posters, so $C(0)=18$ is a bill the shop charges.

What each part of the poster price answers
Part	Value	Unit	Meaning for this shop
Slope	$0.75$	dollars per poster	Cost added for each additional poster
Intercept	$18$	dollars	Cost when the poster count is zero
$C(20)$	$33$	dollars	Cost of an order of 20 posters

The order of $20$ posters costs $0.75(20)+18=15+18=33$ dollars. One more poster changes that bill from $\$33$ to $\$33.75$ . The $\$18$ stays in the bill, and the extra $\$0.75$ is the slope.

Build the same price from two orders

Suppose the only information given is that $12$ posters cost $\$27$ and $36$ posters cost $\$45$ . The slope is the change in cost divided by the change in poster count:

$m=\frac{45-27}{36-12}=\frac{18}{24}=\frac{3}{4}=0.75$

The unit is dollars per poster. Use the $12$ -poster order to find the intercept: $27=0.75(12)+b$ , so $27=9+b$ and $b=18$ . The price is $C(n)=0.75n+18$ .

The second order confirms it: $0.75(36)+18=27+18=45$ . Both orders have to match. An equation that fits the first total and misses the second describes a different price.

A negative slope lowers a starting balance

The same shop sells a drink card with a $\$45$ starting balance. Each drink costs $\$2.75$ , and $2.75=\frac{11}{4}$ . After $d$ drinks, the balance in dollars is

$B(d)=45-2.75d$

The slope is $-2.75$ dollars per drink: each drink lowers the balance by $\$2.75$ . The intercept is $\$45$ , the balance when $d=0$ . After $6$ drinks, $B(6)=45-16.50=28.50$ dollars.

The card pays for a drink when the balance before that drink is at least $\$2.75$ . Since $2.75(16)=44$ , sixteen drinks leave $B(16)=1$ dollar. The next drink costs $\$1.75$ more than that remainder, because $2.75-1=1.75$ . Substituting $17$ gives $B(17)=45-46.75=-1.75$ , the same shortfall written as a negative number. The greatest whole number of drinks the card can pay for is $16$ .

Separate the slope from the average cost

For $20$ posters, the bill is $\$33$ , so the average is $\frac{33}{20}=1.65$ dollars per poster. For $40$ posters, the bill is $0.75(40)+18=48$ dollars, and the average is $\frac{48}{40}=1.20$ dollars per poster.

The slope is still $0.75$ . From $20$ posters to $40$ posters, the bill rises by $48-33=15$ dollars, so the added cost is $\frac{15}{20}=0.75$ dollars per poster. Each average includes a share of the $\$18$ design fee, and that share shrinks as the order grows.

A fixed fee changes the average in the same way in the ratios and unit-rate guide. Use the slope for the cost of one more poster. Use an average when the question asks for the total bill divided by the poster count.

Rewrite a standard-form price

The shop could also write the poster rule as $4C-3n=72$ , with $C$ in dollars and $n$ the number of posters. Add $3n$ to both sides, then divide by $4$ :

$4C=3n+72$

$C=\frac{3}{4}n+18=0.75n+18$

The rewritten equation is the same price: the slope is $\frac{3}{4}$ dollars per poster, and the value at zero posters is $18$ dollars. For any whole-number order in this guide, $4C(n)-3n=72$ . The balance steps are the same operations used in the guide to linear equations with variables on both sides.

A model that adds the same dollar amount for each poster stays linear. A quantity multiplied by the same factor over each equal period follows a different rule, which the exponential growth guide separates into a starting amount and a multiplier.

Try four original practice questions

Use $C(n)=0.75n+18$ for the poster price and $B(d)=45-2.75d$ for the drink card. Exactly one choice answers each question.

In $C(n)=0.75n+18$ , what does $0.75$ mean?
- A. The cost is $\$0.75$ when no posters are ordered.
- B. The cost increases by $\$0.75$ for each additional poster.
- C. The cost increases by $\$18$ for each additional poster.
- D. An order of $0.75$ posters costs $\$18$ .
What is the cost of $48$ posters?
- A. $\$18.75$
- B. $\$36.00$
- C. $\$54.00$
- D. $\$66.00$
Two new orders cost $\$24$ for $8$ posters and $\$39$ for $28$ posters. Which equation matches both orders?
- A. $C(n)=0.75n+18$
- B. $C(n)=0.75n+24$
- C. $C(n)=15n+18$
- D. $C(n)=1.50n+18$
In $B(d)=45-2.75d$ , what does $-2.75$ mean?
- A. The balance increases by $\$2.75$ for each additional drink.
- B. The balance decreases by $\$45$ for each additional drink.
- C. The starting balance is $\$2.75$ .
- D. The balance decreases by $\$2.75$ for each additional drink.

Show answers and checks

Choice B. Each additional poster adds $\$0.75$ , so the slope’s unit is dollars per poster.
- Choice A is the cost at zero posters assigned to the slope. That cost is $C(0)=18$ dollars.
- Choice C gives the design fee the job of the slope. The fee is the constant $18$ .
- Choice D pairs $0.75$ with the poster count and $18$ with the cost. At $n=0.75$ , the model gives $0.75(0.75)+18=18+\frac{9}{16}$ .
Choice C. $C(48)=0.75(48)+18=36+18=54$ , so $48$ posters cost $\$54$ .
- Choice A is $C(1)=18.75$ dollars, the cost of one poster.
- Choice B is $0.75(48)=36$ dollars, the poster charges with the $\$18$ design fee left out.
- Choice D is $48+18=66$ dollars, the poster count added to the design fee.
Choice A. The slope between the orders is $\frac{39-24}{28-8}=\frac{15}{20}=0.75$ . Then $24=0.75(8)+b$ , so $b=18$ . The equation $C(n)=0.75n+18$ also gives $0.75(28)+18=39$ .
- Choice B uses the first order’s $\$24$ total as the intercept. It gives $0.75(8)+24=30$ and $0.75(28)+24=45$ .
- Choice C uses the $\$15$ change in cost as the slope. At $8$ posters it gives $15(8)+18=138$ .
- Choice D uses $\$1.50$ per poster, twice the slope. At $8$ posters it gives $1.50(8)+18=30$ .
Choice D. The coefficient $-2.75$ lowers the balance by $\$2.75$ for each additional drink.
- Choice A reverses the direction. From $d=0$ to $d=1$ , the balance goes from $\$45$ to $\$42.25$ .
- Choice B uses the starting balance as the change per drink. The starting balance is the constant $45$ .
- Choice C reads the drink price as the value at $d=0$ . That value is $B(0)=45$ dollars.

Name the units before you finish

Write the input unit and the output unit beside the equation.
State the change in the output for an increase of 1 in the input, and attach “per” to that rate.
Evaluate the model at 0 and compare the result with the situation’s starting charge.
For a negative slope, check that a larger input produces a smaller output.
Substitute every given order into a finished equation.

If a choice described a different part of the model, record the part you circled and the unit you gave it. The practice error log keeps that note for a later recheck.

Try a free SAT practice question with Kibo and explain what the rate and the starting value mean in your own words.

Prepared with AI assistance. The poster prices, drink-card balances, slopes, intercepts, and every practice choice were checked with separate exact arithmetic. No human review or score improvement is claimed. Sources checked November 13, 2026. SAT is a registered trademark of College Board, which is not affiliated with or endorsing this guide.