Question

Four times a number added to 3 times a larger number is 31. Seven subtracted from the larger number is equal to twice the smaller number. Let x represent the smaller number and y represent the larger number. Which equations represent this situation?
y = negative four-thirds x + 31. y = 2 x + 7.
y = negative four-thirds x + StartFraction 31 Over 3 EndFraction. Y = 2 x + 7.
y = negative four-thirds x + 31. y = negative 2 x + 7.
y = negative four-thirds x + StartFraction 31 Over 3 EndFraction. Y = negative 2 x + 7.

Answer 1

Let's analyze the problem step by step to determine the correct system of equations.

1. **First statement**: "Four times a number added to 3 times a larger number is 31."
- Let \(x\) represent the smaller number and \(y\) represent the larger number.
- This statement translates to the equation:
\[
4x + 3y = 31
\]

2. **Second statement**: "Seven subtracted from the larger number is equal to twice the smaller number."
- This translates to the equation:
\[
y - 7 = 2x
\]
- Rearranging gives:
\[
y = 2x + 7
\]

Now we have the following two equations:
1. \(4x + 3y = 31\)
2. \(y = 2x + 7\)

Next, we need to express the first equation in terms of \(y\):
- From \(4x + 3y = 31\), we can isolate \(y\):
\[
3y = 31 - 4x \quad \Rightarrow \quad y = \frac{31 - 4x}{3} \quad \Rightarrow \quad y = -\frac{4}{3}x + \frac{31}{3}
\]

Thus, the system of equations is:
1. \(y = -\frac{4}{3}x + \frac{31}{3}\)
2. \(y = 2x + 7\)

Now let's compare this with the answer choices provided:

- The correct formulation corresponds to:
\[
y = -\frac{4}{3}x + \frac{31}{3} \quad \text{and} \quad y = 2x + 7
\]

From your options, the correct one is:
**"y = negative four-thirds x + StartFraction 31 Over 3 EndFraction. y = 2 x + 7."**