Given the system:
[tex]\[ y = x^3 - 1 \][/tex]
[tex]\[ y = 7 \][/tex]

Column A: [tex]\(\frac{y}{2}\)[/tex]
Column B: [tex]\(|x|\)[/tex]

A. Column A's quantity is greater.
B. Column B's quantity is greater.
C. The quantities are the same.
D. The relationship cannot be determined from the given information.



Answer :

To compare the quantities in Column A and Column B for the given system of equations, let’s work through the problem step-by-step.

### Step 1: Solve for [tex]\( x \)[/tex]
We are given the system of equations:
[tex]\[ y = x^3 - 1 \][/tex]
[tex]\[ y = 7 \][/tex]

First, we equate [tex]\( y \)[/tex] to find the value of [tex]\( x \)[/tex]:
[tex]\[ 7 = x^3 - 1 \][/tex]

Add 1 to both sides to isolate the [tex]\( x^3 \)[/tex] term:
[tex]\[ 8 = x^3 \][/tex]

Taking the cube root of both sides:
[tex]\[ x = \sqrt[3]{8} \][/tex]
[tex]\[ x = 2 \][/tex]

### Step 2: Determine Column A and Column B
Now that we have the value of [tex]\( x = 2 \)[/tex], we can calculate the quantities in Column A and Column B.

Column A:
[tex]\[ \frac{y}{2} \][/tex]
Since [tex]\( y = 7 \)[/tex]:
[tex]\[ \frac{7}{2} = 3.5 \][/tex]

Column B:
[tex]\[ |x| \][/tex]
Since [tex]\( x = 2 \)[/tex]:
[tex]\[ |2| = 2 \][/tex]

### Step 3: Compare the Quantities
Finally, we compare the quantities in Column A and Column B:
[tex]\[ \text{Column A: } 3.5 \][/tex]
[tex]\[ \text{Column B: } 2 \][/tex]

Since 3.5 is greater than 2, we conclude that:
[tex]\[ \text{Column A's quantity is greater.} \][/tex]

Thus, the final answer is: Column A's quantity is greater.