Expressions and Equations: Analyzing Relationships Between Dependent and Independent Variables (CCSS.6.EE.9)
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Common Core 6th Grade Math › Expressions and Equations: Analyzing Relationships Between Dependent and Independent Variables (CCSS.6.EE.9)
A music streaming app charges 10 dollars per month plus 3 dollars for each premium song download. Let $x$ be the number of downloads in a month and $y$ be the total monthly cost in dollars. Which equation represents the relationship?
$y=3x+10$
$y=10x+3$
$x=3y+10$
$y=3x$
Explanation
$y$ depends on $x$: each download adds 3 dollars (slope 3), and there is a 10-dollar starting fee (intercept 10), so $y=3x+10$. The other choices mix up the fee/rate, reverse variables, or ignore the fee.
Maya has 5 dollars saved and earns 2 dollars for each cup of lemonade she sells. Let $x$ be the number of cups sold and $y$ be the total amount of money she has. What is the value of $y$ when $x=7$?
14
17
19
9
Explanation
The relationship is $y=2x+5$. Substitute $x=7$: $y=2(7)+5=14+5=19$. Here, 2 is the rate per cup (slope) and 5 is the starting amount (intercept).
A cyclist travels at a constant speed of 12 miles/hour. Let $x$ be time in hours and $y$ be distance in miles. Which equation represents the relationship?
$y=x+12$
$x=12y$
$y=\frac{12}{x}$
$y=12x$
Explanation
Distance depends on time at a constant rate: $y=12x$. The slope 12 is miles per hour; there is no starting distance, so the intercept is 0.
A community center charges a 20 dollar sign-up fee plus 5 dollars for each fitness class attended. Let $x$ be the number of classes and $y$ be the total cost in dollars. What is the value of $y$ when $x=6$?
30
50
31
106
Explanation
The relationship is $y=5x+20$. Substitute $x=6$: $y=5(6)+20=30+20=50$. The slope 5 is the cost per class; the intercept 20 is the sign-up fee.
A lemonade stand charges a setup fee of 3 dollars plus 2 dollars for each cup sold. Let $x$ be the number of cups and $y$ be the total cost in dollars.
Which equation represents the relationship?
$y = 2x + 3$
$y = 3x + 2$
$x = 2y + 3$
$y = 2x - 3$
Explanation
$y$ depends on $x$. The slope (2) is the cost per cup, and the intercept (3) is the initial fee, so $y = 2x + 3$.
A runner moves at a constant speed of 5 miles per hour. Let $x$ be the time in hours and let $y$ be the distance in miles. Some pairs are $(1,5)$, $(2,10)$, $(3,15)$.
What is the value of $y$ when $x = 4$?
5
9
20
4
Explanation
At constant speed, $y = 5x$. Substituting $x=4$ gives $y = 5(4) = 20$ miles.
An online game charges a membership fee of 10 dollars plus 4 dollars for each add-on pack. Let $x$ be the number of packs and $y$ be the total cost in dollars.
Which equation represents the relationship?
$y = 10x + 4$
$y = 4x - 10$
$x = 4y + 10$
$y = 4x + 10$
Explanation
$y$ depends on $x$. The slope (4) is cost per pack and the intercept (10) is the membership fee, so $y = 4x + 10$.
A ride service charges 3 dollars to start plus 2 dollars for each mile. Let $x$ be the miles traveled and let $y$ be the total cost in dollars. Which equation represents the relationship?
$y = 2x + 3$
$y = 3x + 2$
$y = 2x - 3$
$x = 2y + 3$
Explanation
$y$ depends on $x$. The slope (2) is the cost per mile, and the intercept (3) is the starting fee, so $y = 2x + 3$. The other choices mix up the fee and rate, use the wrong sign, or swap $x$ and $y$.
A plant is 5 centimeters tall and grows 2 centimeters each day. Let $x$ be the number of days and $y$ be the plant's height in centimeters. What is the value of $y$ when $x=7$?
14
17
19
9
Explanation
The relationship is $y = 2x + 5$ (2 cm per day, starting height 5). Substitute $x=7$: $y = 2(7) + 5 = 14 + 5 = 19$. The distractors come from forgetting the starting height (14), adding incorrectly (17), or subtracting instead (9).
A party store charges a 4 dollar delivery fee plus 1 dollar for each balloon. Let $x$ be the number of balloons and $y$ be the total cost in dollars. Which equation represents the relationship?
$y = 4x + 1$
$y = x + 4$
$y = x - 4$
$x = y + 4$
Explanation
Each balloon adds 1 dollar (slope 1), and the delivery fee is 4 dollars (intercept 4), so $y = x + 4$. Other options reverse numbers, subtract the fee, or swap variables.