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?

Question

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."