How much larger is `13x^(2) - 7 y^(2) " than " 6x^(2) - 9y^(2)`?

Options

A. `7x^(2) - 2y^(2)`
B. `7x^(2) + 2y^(2)`
C. `6x^(2) - 3y^(2)`
D. `6x^(2) + 3y^(2)`

Answer

To find out how much larger the expression \( 13x^2 - 7y^2 \) is than \( 6x^2 - 9y^2 \), we can follow these steps:

### Step 1: Identify the expressions
Let:
- \( a = 13x^2 - 7y^2 \)
- \( b = 6x^2 - 9y^2 \)

### Step 2: Set up the subtraction
To find out how much larger \( a \) is than \( b \), we need to calculate \( a - b \):
\[ a - b = (13x^2 - 7y^2) - (6x^2 - 9y^2) \]

### Step 3: Distribute the negative sign
Now, distribute the negative sign across the second expression:
\[ a - b = 13x^2 - 7y^2 - 6x^2 + 9y^2 \]

### Step 4: Combine like terms
Next, combine the like terms:
- For \( x^2 \) terms: \( 13x^2 - 6x^2 = 7x^2 \)
- For \( y^2 \) terms: \( -7y^2 + 9y^2 = 2y^2 \)

Putting it all together:
\[ a - b = 7x^2 + 2y^2 \]

### Final Result
Thus, \( 13x^2 - 7y^2 \) is larger than \( 6x^2 - 9y^2 \) by:
\[ 7x^2 + 2y^2 \]

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