Question

When solving an equation, Bianca's first step is shown below. Which property justifies Bianca's first step?
Original Equation:
Original Equation:
9, left parenthesis, 7, x, squared, plus, 2, right parenthesis, plus, 90, equals, minus, 63, x, squared, minus, 72
9(7x
2
+2)+90=
−63x
2
−72
First Step:
First Step:
left parenthesis, 7, x, squared, plus, 2, right parenthesis, plus, 10, equals, minus, 7, x, squared, minus, 8
(7x
2
+2)+10=
−7x
2
−8

Answer 1

To determine the property that justifies Bianca's first step, let's analyze what she did.

The original equation is:

\[ 9(7x^2 + 2) + 90 = -63x^2 - 72 \]

In her first step, she seems to have factored out or simplified terms from both sides of the equation. She rewrote the equation as:

\[ (7x^2 + 2) + 10 = -7x^2 - 8 \]

It appears that she distributed the 9 from the left side of the original equation to the terms inside the parentheses, leading to:

\[ 9 \cdot 7x^2 + 9 \cdot 2 + 90 = -63x^2 - 72\]

Calculating \(9 \cdot 7x^2\) gives \(63x^2\), and calculating \(9 \cdot 2\) gives \(18\). Therefore, the left side simplifies to:

\[ 63x^2 + 18 + 90 = -63x^2 - 72\]

Now, combining like terms on the left side gives:

\[ 63x^2 + 108 = -63x^2 - 72 \]

So the next step is to move all terms involving \(x^2\) to one side and constant terms to the other.

However, Bianca's representation looks like a simplification based on balancing both sides and, possibly, a reorganization.

The property that justifies Bianca's first step is the **Distributive Property**, which allows the distribution of multiplication over addition.

Thus, the correct justification is:

**Distributive Property**.