Creating Equations: Creating and Graphing Equations in Two or More Variables (CCSS.A-CED.2)
Common Core High School Algebra · Learn by Concept
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Common Core High School Algebra › Creating Equations: Creating and Graphing Equations in Two or More Variables (CCSS.A-CED.2)
A bike shop charges a flat rental fee of 8 dollars plus 3 dollars for each hour you keep the bike. Let $C$ be the total cost in dollars and $h$ be the number of hours. Which equation represents the relationship?
$C = 8 + 3h$
$C = 3 + 8h$
$C = 8 + \frac{h}{3}$
$C = 11 + 3h$
Explanation
The cost starts at 8 dollars (the $y$-intercept) and increases by 3 dollars per hour (the slope), so $C = 8 + 3h$. Choice B swaps the rate and the fee. Choice C divides by 3 instead of multiplying, misinterpreting the rate. Choice D uses the wrong starting fee.
A temperature in Celsius is converted to Fahrenheit by a linear rule. Freezing occurs at 0 Celsius which is 32 Fahrenheit, and each 1-degree Celsius increase corresponds to a 1.8-degree Fahrenheit increase. Let $F$ be the temperature in Fahrenheit when the Celsius temperature is $C$. Which equation represents the relationship?
$F = \tfrac{5}{9}C + 32$
$F = \tfrac{9}{5}(C + 32)$
$F = 32C + \tfrac{9}{5}$
$F = \tfrac{9}{5}C + 32$
Explanation
The slope is $\tfrac{9}{5}=1.8$ degrees Fahrenheit per degree Celsius and the $y$-intercept is 32, so $F = \tfrac{9}{5}C + 32$. Choice A uses the reciprocal slope. Choice B incorrectly adds 32 inside the multiplication. Choice C swaps the roles of slope and intercept.
A tank initially contains 150 gallons of water and is drained at a constant rate of 2 gallons per minute. Let $V$ be the volume in gallons after $t$ minutes. Which equation represents the relationship?
$V = 2t + 150$
$V = 150 - 2t$
$V = 150t - 2$
$V = 150 - \tfrac{t}{2}$
Explanation
The starting amount is 150 gallons (intercept), and the volume decreases by 2 gallons each minute (slope $-2$), so $V = 150 - 2t$. Choice A shows an increase instead of a decrease. Choice C swaps intercept and rate. Choice D divides by 2 instead of subtracting 2 per minute.
A taxi charges a base fare of 3 dollars plus 1.75 dollars for each mile driven. Let $c$ be the total cost in dollars and $m$ be the number of miles. Which equation represents the relationship?
$c = 1.75 + 3m$
$c = 3 + \tfrac{m}{1.75}$
$c = 3 + 1.75m$
$c = 4 + 1.75m$
Explanation
The cost starts at 3 dollars (intercept) and increases by 1.75 dollars per mile (slope), so $c = 3 + 1.75m$. Choice A swaps the rate and the base. Choice B divides by 1.75, misapplying the per-mile rate. Choice D changes the base fare to 4 dollars.
Before starting a shift, a babysitter has already received 25 dollars in tips. During the shift, she earns 14 dollars per hour in wages. Let $M$ be her total money after $h$ hours. Which equation represents the relationship?
$M = 25 + 14h$
$M = 14 + 25h$
$M = 25 + \tfrac{h}{14}$
$M = 23 + 14h$
Explanation
She starts with 25 dollars (intercept) and adds 14 dollars for each hour worked (slope), giving $M = 25 + 14h$. Choice B swaps the intercept and slope. Choice C divides by 14 instead of multiplying. Choice D uses the wrong starting amount.
A bike shop charges a start fee of 5 dollars plus 7 dollars per hour to rent a bike. Let $x$ be the number of hours and let $y$ be the total cost in dollars. Which equation represents the relationship?
$y=7+5x$
$y=5+7x$
$y=12x$
$y=5x+6$
Explanation
The rate is 7 dollars per hour, so the coefficient of $x$ is 7; the start fee of 5 dollars is the $y$-intercept. Thus $y=5+7x$. Choice A swaps the slope and intercept, C treats the fees as a single per-hour rate and drops the fixed fee, and D makes an arithmetic slip in the hourly rate.
Water temperature in degrees Fahrenheit increases by 1.8 for each 1 degree Celsius, and when it is 0 degrees Celsius it is 32 degrees Fahrenheit. Let $x$ be the temperature in degrees Celsius and let $y$ be the temperature in degrees Fahrenheit. Which equation represents the relationship?
$y=32x+1.8$
$y=x+32$
$y=1.8x+30$
$y=1.8x+32$
Explanation
The slope is 1.8 Fahrenheit per 1 Celsius, and the $y$-intercept is 32 at $x=0$, so $y=1.8x+32$. Choice A swaps the slope and intercept, B assumes a slope of 1, and C makes an intercept arithmetic slip (should be 32, not 30).
A streaming service charges a monthly base fee of 9 dollars plus 2 dollars for each movie rental. Let $x$ be the number of movie rentals in a month and let $y$ be the total monthly cost in dollars. Which equation represents the relationship?
$y=9+2x$
$y=2+9x$
$y=11x$
$y=9+3x$
Explanation
The fixed fee of 9 dollars is the $y$-intercept, and the cost increases by 2 dollars per rental, so the slope is 2: $y=9+2x$. Choice B swaps slope and intercept, C treats the fixed fee as if it were per rental, and D uses the wrong per-rental rate.
A tank contains 500 liters of water and is draining at a constant rate of 6 liters per minute. Let $x$ be the time in minutes and let $y$ be the amount of water in liters. Which equation represents the relationship?
$y=6x+500$
$y=500+6x$
$y=500-6x$
$y=494-6x$
Explanation
The tank starts at 500 liters (the $y$-intercept) and decreases by 6 liters per minute, so the slope is $-6$: $y=500-6x$. Choices A and B show an increase instead of a decrease, and D makes an intercept arithmetic slip (should start at 500, not 494).
A club pays a setup fee of 40 dollars to print T-shirts plus 8 dollars for each shirt. Let $x$ be the number of shirts and let $y$ be the total cost in dollars. Which equation represents the relationship?
$y=8+40x$
$y=48x$
$y=40+8x$
$y=40+7x$
Explanation
The setup fee of 40 dollars is the $y$-intercept, and the cost increases by 8 dollars per shirt, so the slope is 8: $y=40+8x$. Choice A swaps slope and intercept, B treats the setup fee as if it were per shirt and drops the fixed fee, and D uses the wrong per-shirt rate.