Work through the questions first, then open the answer key and explanations to check your work.
A school theater club is preparing gift bags for a fundraiser. Each snack bag contains 3 cookies, and each drink bag contains 5 juice boxes. Let \(x\) represent the number of snack bags and \(y\) represent the number of drink bags. The total number of items is modeled by the equation \(3x+5y=79\).
What does the ordered pair \((18,5)\) represent in this context?
A community center sells smoothie cups and fruit bowls during a fundraiser. Each smoothie cup costs $6, and each fruit bowl costs $4. The center earned a total of $92.
If \(s\) represents the number of smoothie cups sold and \(f\) represents the number of fruit bowls sold, which equation models this situation?
A community theater sold standard tickets for $12 each and student tickets for $7 each for a weekend show. The theater collected a total of $263 from ticket sales. If \(s\) represents the number of standard tickets sold and \(t\) represents the number of student tickets sold, which equation represents this situation?
A school club sold notebooks for \($4\) each and water bottles for \($9\) each at a fundraiser. The club sold 7 water bottles and collected a total of \($131\). How many notebooks did the club sell?
A school library received a shipment of 87 new books. Of those books, 34 are science books and the rest are history books. How many history books did the library receive?
A line in the xy-plane passes through the points \((-4,7)\) and \((6,-8)\). What is the slope of the line?
A line in the xy-plane passes through the points \((-4, 7)\) and \((6, -8)\). What is the slope of the line?
A catering company packs meals for an event. Each large tray weighs \(x\) pounds, and each dessert box weighs \(y\) pounds. The total weight of 5 large trays and 2 dessert boxes is represented by the equation \(5x+2y=74\).
Which choice best interprets the value 74 in this context?
The graph shows a line passing through the points \((0,9)\) and \((3,0)\). Let \(P\) represent a positive constant.
Which equation represents the line?
The graph shows a line in the xy-plane passing through the points \((0,9)\), \((1,6)\), and \((3,0)\). Let \(P\) represent a positive constant.
Which equation represents the line?
In the xy-plane, the graph of the equation \(\frac{3x}{5}-\frac{y}{4}=7-\frac{x}{10}\) intersects the y-axis at a point. What is the y-coordinate of that point?
In the xy-plane, the graph of the equation \(\frac{3x}{4}-\frac{5y}{6}=7\) crosses the y-axis at \((0,k)\). What is the value of \(k\)?
There are 18 snack bags and 5 drink bags.
In the equation, \(x\) represents the number of snack bags and \(y\) represents the number of drink bags. The ordered pair \((18,5)\) means \(x=18\) and \(y=5\). Therefore, there are 18 snack bags and 5 drink bags. Substituting into the equation confirms the pair is a solution: \(3(18)+5(5)=54+25=79\).
\(6s+4f=92\)
Each smoothie cup contributes \(6\) dollars, so the total earned from smoothie cups is \(6s\). Each fruit bowl contributes \(4\) dollars, so the total earned from fruit bowls is \(4f\). Since the combined earnings were \(92\) dollars, the correct equation is \(6s+4f=92\).
\(12s+7t=263\)
Each standard ticket costs $12, so the money earned from standard tickets is \(12s\). Each student ticket costs $7, so the money earned from student tickets is \(7t\). The total amount collected was $263, so the two amounts are added together to make the equation \(12s+7t=263\).
17
Let \(x\) represent the number of notebooks sold. The 7 water bottles brought in \(9 \times 7 = 63\) dollars. Since the total amount collected was \(131\) dollars, the notebooks brought in \(131 - 63 = 68\) dollars. Each notebook costs \(4\) dollars, so solve \(4x = 68\). Dividing by 4 gives \(x = 17\). Therefore, the club sold 17 notebooks.
Let \(x\) represent the number of history books. The total number of books is the sum of the science books and the history books, so the equation is \(x+34=87\). Subtract 34 from both sides:
\(x=87-34=53\)
The library received 53 history books.
Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
Substitute the coordinates:
\(m=\frac{-8-7}{6-(-4)}=\frac{-15}{10}\)
Simplify the fraction:
\(m=\frac{-15}{10}=-\frac{3}{2}\)
So the slope of the line is \(-\frac{3}{2}\), which is also equal to \(-1.5\).
Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
Substitute the coordinates:
\(m=\frac{-8-7}{6-(-4)}=\frac{-15}{10}\)
Simplify the fraction:
\(\frac{-15}{10}=\frac{-3}{2}\)
So, the slope of the line is \(-\frac{3}{2}\), which is also equal to \(-1.5\).
The total weight of 5 large trays and 2 dessert boxes combined, in pounds
The term \(5x\) represents the combined weight of 5 large trays, and the term \(2y\) represents the combined weight of 2 dessert boxes. Since these quantities are added together and equal 74, the value 74 represents the total weight of all 5 trays and 2 dessert boxes combined.
\(6x+Py=18\)
Using the two points \((0,9)\) and \((3,0)\), the slope is \(\frac{0-9}{3-0}=-3\). The y-intercept is 9, so the equation in slope-intercept form is \(y=-3x+9\).
Rewriting in standard form gives \(3x+y=9\). Multiplying every term by 2 gives the equivalent equation \(6x+2y=18\). Since \(P\) is positive and replaces the coefficient of \(y\), \(P=2\), so the correct equation is \(6x+Py=18\).
\(6x+Py=18\)
Using the points \((0,9)\) and \((1,6)\), the slope is \(\frac{6-9}{1-0}=-3\). Since the line crosses the y-axis at \(9\), the equation in slope-intercept form is \(y=-3x+9\).
Rewrite in standard form:
\(3x+y=9\)
Multiplying both sides by 2 gives:
\(6x+2y=18\)
Because \(P\) is a positive constant replacing the coefficient of \(y\), \(P=2\). Therefore, the correct equation is \(6x+Py=18\).
A point on the y-axis has \(x=0\). Substitute 0 for \(x\) in the equation:
\(\frac{3(0)}{5}-\frac{y}{4}=7-\frac{0}{10}\)
This simplifies to:
\(-\frac{y}{4}=7\)
Multiply both sides by 4:
\(-y=28\)
So, \(y=-28\). Therefore, the y-coordinate of the y-intercept is \(-28\).
A point on the y-axis has an x-coordinate of 0, so substitute \(x=0\) into the equation:
\(\frac{3(0)}{4}-\frac{5y}{6}=7\)
This simplifies to:
\(-\frac{5y}{6}=7\)
Multiply both sides by 6:
\(-5y=42\)
Then divide by \(-5\):
\(y=-\frac{42}{5}\)
Therefore, \(k=-\frac{42}{5}\), which is also equal to \(-8.4\).
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