Question

travis purchases 15 fish for a large aquarium. he purchases goldfish for $2 each and guppies for $3 each. travis spends $39 on the fish. a graph of two linear equations representing the number of each type of fish travis could purchase is shown. what does the intersection of the two linear equations represent? a travis purchases 6 guppies. b travis purchases 6 goldfish. c travis spends a total of $6 on guppies. d travis spends a total of $6 on goldfish.

Explanation:

Step1: Understand the Graph Axes

The x - axis is Goldfish (number), y - axis is Guppies (number). Intersection point has coordinates (let's assume from graph) where x (goldfish) and y (guppies) are values. But also, we know two equations: \(x + y=15\) (total fish) and \(2x + 3y = 39\) (total cost, goldfish \(\$2\), guppies \(\$3\)).

Step2: Solve the System of Equations

Solve \(

$$\begin{cases}x + y=15\\2x + 3y=39\end{cases}$$

\)
From first equation, \(x = 15 - y\). Substitute into second: \(2(15 - y)+3y=39\)
\(30-2y + 3y=39\)
\(y = 9\), then \(x=15 - 9 = 6\)? Wait, no, wait. Wait, goldfish are \(x\) (x - axis), guppies \(y\) (y - axis). Wait, maybe I mixed up. Wait, the problem says: Travis purchases 15 fish (so \(x + y=15\), x = goldfish, y = guppies), and spends \(\$39\) ( \(2x+3y = 39\) ). Solving:
From \(x + y=15\), \(x = 15 - y\). Substitute into \(2x + 3y=39\):
\(2(15 - y)+3y=39\)
\(30-2y + 3y=39\)
\(y = 9\), \(x = 6\). Wait, but the intersection point: let's check the options. Option a: Travis purchases 6 guppies? No, x is goldfish (6), y is guppies (9). Wait, no, maybe the axes are reversed? Wait, the graph: x - axis is Goldfish (labeled Goldfish), y - axis is Guppies (labeled Guppies). The intersection point: let's see the graph, the two lines intersect at (6,9)? Wait, no, maybe I made a mistake. Wait, the options: a) 6 guppies, b) 6 goldfish, c) \$6 on guppies, d) \$6 on goldfish.

Wait, let's re - evaluate. The two equations: one is the total number of fish \(x + y=15\) (x = goldfish, y = guppies), the other is total cost \(2x+3y = 39\). Solving:

Multiply first equation by 2: \(2x + 2y=30\)

Subtract from second equation: \((2x + 3y)-(2x + 2y)=39 - 30\)

\(y = 9\), then \(x=15 - 9 = 6\). So x (goldfish) is 6, y (guppies) is 9. So the intersection represents the solution to both equations, i.e., Travis purchases 6 goldfish (x = 6) and 9 guppies. So option b: Travis purchases 6 goldfish. Wait, no, option b is "Travis purchases 6 goldfish", option a is 6 guppies. So the intersection point's x - coordinate is 6 (goldfish), y - coordinate is 9 (guppies). So the intersection represents the number of goldfish and guppies that satisfy both total fish and total cost. So the correct option is b? Wait, no, wait the options:

a) Travis purchases 6 guppies. (y = 6? No, our solution is y = 9)

b) Travis purchases 6 goldfish. (x = 6, which matches our solution)

c) Travis spends a total of \$6 on guppies. (3y = 6 → y = 2, not our solution)

d) Travis spends a total of \$6 on goldfish. (2x = 6 → x = 3, not our solution)

So the intersection point is where x = 6 (goldfish) and y = 9 (guppies), so it represents Travis purchasing 6 goldfish (option b) and 9 guppies, satisfying both total fish (6 + 9 = 15) and total cost (26+39 = 12 + 27 = 39).

Answer:

b. Travis purchases 6 goldfish.