Question

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?
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 \( x - 3y = -9 \).