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.
