16. Graphs of equations with three variables Transcript
Antonio is buying salad and pizza for a company lunch. Suppose that a bowl of salad costs $5.00, and a slice of pizza costs $3.00.
Let E be the amount in dollars that Antonio spends on salad and pizza. If Antonio buys S bowls of salad and P slices of pizza, then the total amount of money he spends ( E ) can be represented by the equation E=$5.00S+$3.00P Correct .
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Every bowl of salad Antonio buys costs him $5.00. Therefore, if he buys S bowls of salad, the amount he spends on salad is $5.00S .
Similarly, every slice of pizza Antonio buys costs him $3.00. Therefore, if he buys P slices of pizza, the amount he spends on pizza is $3.00P .
If Antonio spends $5.00S on salad and $3.00P on pizza, his total expenditure is E=$5.00S+$3.00P .
Now rearrange the equation you wrote above so that P is written in terms of E and S . The quantity of pizza he buys can be represented by the equation P=E−($5.00×S)$3.00 Correct .
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You should have found before that if Antonio spends $5.00S on salad and $3.00P on pizza, then his total expenditure is E=($5.00×S)+($3.00×P) .
Solving this for P yields:
E = $5.00S+$3.00P
E−($5.00×S) = $3.00P
P = E−$5.00S$3.00
Suppose Antonio has $30.00 to spend on salad and pizza; that is, E=$30.00 .
Complete the following table with the values of S or P that make the equation true.
Hint: To complete the first row, determine the number of pizza slices Antonio can purchase with $30.00, when the number of salad bowls he purchases is 0.
Budget
Salad
Pizza
(Dollars)
(Bowls)
(Slices)
30.00 0 10Correct
30.00 3 5Correct
30.00 6Correct 0
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Use the black line (plus symbols) to plot the line illustrating the combinations of salad and pizza that Antonio can purchase with a budget of $30.00.
Your Answer
Initial Budget
New Budget
0
2
4
6
8
10
12
14
14
12
10
8
6
4
2
0
PIZZA (Slices)
SALAD (Bowls)
Correct Answer
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You determined above that the relationship between the three variables could be written as P=E−$5.00S$3.00 . For a given value of E , you have a direct relationship between S and P , P=$30.00−$5.00S$3.00 .
You can, therefore, compute values of P by plugging in values of S . For instance, when S is 0, P is computed as follows:
P = $30.00−($5.00×0)$3.00
P = $30.00$3.00
P = 10
Similarly, plugging in 3 for S yields P=5 , and plugging in 0 for P yields S=6 .
Each row represents the coordinates to plot the line representing affordable combinations with a budget of $30.00. Thus the $30.00 budget line is defined by the endpoints (0,10) and (6,0), and the point (3, 5) lies on this line as well.
Now suppose Antonio’s income is cut in half : That is, he has 50% less money to spend than he did before.
On the previous graph, use the blue line (circle symbols) to plot Antonio's new budget constraint.
From the calculations above, you know that the general formula for Antonio’s budget constraint may be written as follows:
P = E−($5.00S)$3.00
If Antonio’s income were cut in half, from E to 0.5E , this would change to the following (where E=$30.00 is the initial amount of money that Antonio had):
P = 0.5×$30.00−($5.00S)$3.00
You can, therefore, compute values of P by plugging in values of S . For instance, when S is 0, P is computed as follows:
P = 0.5×$30.00−($5.00×0)$3.00
P = 0.5×$30.00$3.00
P = 5
This tells you that the new budget constraint has an endpoint of (0, 5). To compute the other endpoint, you can solve for S when P is equal to 0, which yields the endpoint (3, 0).
Which of the following statements best summarizes the pattern of causality captured by the graph above? Check all that apply.

Correct A change in S causes a change in P .
Correct A change in E causes a shift of the budget constraint.
Correct The relationship between S and P is not a causal relationship.
Correct A change in P causes a change in S .
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While the graph appears to show a “relationship” between S and P, that relationship is not causal in nature. The line shows how much pizza and salad Antonio can afford. The amount of salad Antonio can afford is a function of his income and the price of salad; the amount of pizza Antonio can afford is a function of his income and the price of pizza. In other words, the amount of salad Antonio can afford is not a function of the amount of pizza he can afford.
The only causation in this problem comes from Antonio’s income: In particular, a change in Antonio’s income alters the set of combinations of goods he can afford.