Question

a canoe travels 3 miles per hour downstream and 1 mile per hour upstream. let x represent the canoe’s speed with no water current (in still water) and y represent the speed of the water current, in miles per hour. then the situation can be represented by this system of equations: x + y = 3, x - y = 1. choose the two correct options. the speed of the canoe in still water is 5 miles per hour. the speed of the water current is 4 miles per hour. the speed of the water current is 5 miles per hour. the speed of the canoe in still water is 4 miles per hour.

Explanation:

Step1: Solve the system of equations

We have the system:

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Add the two equations together:
\((x + y)+(x - y)=9 + 1\)
Simplify the left - hand side: \(x + y+x - y = 2x\), and the right - hand side is \(10\). So \(2x=10\), then \(x = 5\).

Substitute \(x = 5\) into the first equation \(x + y=9\), we get \(5 + y=9\), then \(y=9 - 5=4\)? Wait, no, wait. Wait, if \(x - y=1\) and \(x = 5\), then \(5-y = 1\), so \(y=5 - 1=4\)? Wait, no, let's check again.

Wait, when we add the two equations:

\(x + y+(x - y)=9 + 1\)

\(x + y+x - y=10\)

\(2x=10\), so \(x = 5\). Then substitute \(x = 5\) into \(x - y=1\), we have \(5-y = 1\), so \(y=5 - 1 = 4\)? Wait, but let's check the first equation: \(x + y=5 + 4=9\), which matches. Wait, but let's check the options:

Option 1: "The speed of the canoe in still water is 5 miles per hour" - since \(x = 5\), this is correct.

Option 2: "The speed of the water current is 4 miles per hour" - since \(y = 4\)? Wait, no, wait, if \(x=5\) and \(x - y=1\), then \(y=x - 1=5 - 1 = 4\)? Wait, but let's check the first equation \(x + y=5+4 = 9\), which is correct. Wait, but the third option says "The speed of the water current is 5 miles per hour" which is wrong. The fourth option says "The speed of the canoe in still water is 4 miles per hour" which is wrong. Wait, but let's re - solve the system.

Wait, the two equations are \(x + y=9\) (downstream speed: canoe speed in still water plus current speed) and \(x - y=1\) (upstream speed: canoe speed in still water minus current speed).

Adding the two equations:

\(x + y+x - y=9 + 1\)

\(2x=10\)

\(x = 5\)

Substitute \(x = 5\) into \(x + y=9\), we get \(5 + y=9\), so \(y = 4\)? Wait, no, if \(x - y=1\) and \(x = 5\), then \(5-y = 1\), so \(y=4\). Wait, but let's check the upstream speed: \(x - y=5 - 4 = 1\), which matches. The downstream speed: \(x + y=5 + 4=9\), which matches.

Wait, but the options:

1. "The speed of the canoe in still water is 5 miles per hour" - correct, since \(x = 5\).

1. "The speed of the water current is 4 miles per hour" - correct, since \(y = 4\).

1. "The speed of the water current is 5 miles per hour" - wrong, because \(y = 4\).

1. "The speed of the canoe in still water is 4 miles per hour" - wrong, because \(x = 5\).

Wait, but let's check the calculation of \(y\) again. From \(x - y=1\) and \(x = 5\), we have \(y=x - 1=5 - 1 = 4\). And from \(x + y=9\), \(y=9 - x=9 - 5 = 4\). So \(y = 4\).

So the correct options are:

"The speed of the canoe in still water is 5 miles per hour" and "The speed of the water current is 4 miles per hour"

Answer:

A. The speed of the canoe in still water is 5 miles per hour, B. The speed of the water current is 4 miles per hour