A number, y, is equal to the difference of a larger number and 3. The same number is one-third of the sum of the larger number and 9. Which equations represent the situation?

Question

x + y = 3 and x minus 3 y = negative 27
x + y = 3 and x minus 3 y = negative 9
x minus y = 3 and x minus 3 y = negative 27
x minus y = 3 and x minus 3 y = negative 9

Answer 1

Let's break down the problem step by step to form the correct equations.

1. Understanding the Variables:

   • Let \( x \) be the larger number.
   • Let \( y \) be the number that is defined in relation to \( x \).

2. First Condition:

   • "A number, \( y \), is equal to the difference of a larger number and 3."
   • This can be expressed as: \[ y = x - 3 \]

3. Second Condition:

   • "The same number is one-third of the sum of the larger number and 9."
   • This can be expressed as: \[ y = \frac{1}{3}(x + 9) \]

Now we have the following equations based on the above:

1. \( y = x - 3 \) (Equation 1)
2. \( y = \frac{1}{3}(x + 9) \) (Equation 2)

Next, we can substitute the value of \( y \) from Equation 1 into Equation 2 to form a system of equations. For clarity in standard form, we will rewrite both equations to eliminate \( y \):

From Equation 2: \[ 3y = x + 9 \]

Substituting \( y = x - 3 \) into \( 3y = x + 9 \): \[ 3(x - 3) = x + 9 \] \[ 3x - 9 = x + 9 \] \[ 3x - x = 9 + 9 \] \[ 2x = 18 \] \[ x = 9 \]

Now substituting back to find \( y \): \[ y = x - 3 = 9 - 3 = 6 \]

Now we can express everything in the proper format for the choices. Since we simply need to rearrange the equations.

1. From \( y = x - 3 \): \[ x - y = 3 \]

2. From \( y = \frac{1}{3}(x + 9) \) it can be rearranged as: \[ 3y = x + 9 \implies x - 3y = -9 \]

So the correct equations that represent the situation are: \[ x - y = 3 \quad \text{and} \quad x - 3y = -9 \]

Thus, the answer is:

• \( x - y = 3 \) and \( x - 3y = -9 \) which corresponds to the second option:
• \( x - y = 3 \) and \( x - 3y = -9 \).