Question

When solving an equation, Emily's first step is shown below. Which property justifies Emily's first step?
Original Equation:
Original Equation:
minus, 10, left parenthesis, 3, x, squared, minus, 7, right parenthesis, minus, 2, equals, minus, 3, x, squared, plus, 2
−10(3x
2
−7)−2=
−3x
2
+2
First Step:
First Step:
left parenthesis, minus, 30, x, squared, plus, 70, right parenthesis, minus, 2, equals, minus, 3, x, squared, plus, 2
(−30x
2
+70)−2=
−3x
2
+2

Answer 1

Emily's first step involves distributing the \(-10\) across the expression \((3x^2 - 7)\). This is an application of the **Distributive Property**.

The Distributive Property states that for any real numbers \(a\), \(b\), and \(c\):

\[
a(b + c) = ab + ac
\]

In this case, Emily is applying it in the form of:

\[
-10(3x^2 - 7) = -10 \cdot 3x^2 + (-10 \cdot -7) = -30x^2 + 70
\]

Thus, the justification for Emily's first step is the **Distributive Property**.