Question
When solving an equation, Emily's first step is shown below. Which property justifies Emily's first step?
Original Equation:
minus, 8, left parenthesis, minus, x, squared, plus, 7, right parenthesis, plus, 6, equals, minus, 5, x, squared, plus, 4
−8(−x
2
+7)+6=
−5x
2
+4
First Step:
First Step:
minus, 8, left parenthesis, minus, x, squared, plus, 7, right parenthesis, equals, minus, 5, x, squared, minus, 2
−8(−x
2
+7)=
−5x
2
−2
Original Equation:
minus, 8, left parenthesis, minus, x, squared, plus, 7, right parenthesis, plus, 6, equals, minus, 5, x, squared, plus, 4
−8(−x
2
+7)+6=
−5x
2
+4
First Step:
First Step:
minus, 8, left parenthesis, minus, x, squared, plus, 7, right parenthesis, equals, minus, 5, x, squared, minus, 2
−8(−x
2
+7)=
−5x
2
−2
Answers
Answered by
GPT-4o mini
To determine which property justifies Emily's first step in the equation, we need to analyze what she did.
The original equation is:
\[
-8(-x^2 + 7) + 6 = -5x^2 + 4
\]
In the first step, Emily simplified the left side of the equation:
\[
-8(-x^2 + 7) = -5x^2 - 2
\]
Here, Emily applied the **Distributive Property**. The Distributive Property states that for any numbers \(a\), \(b\), and \(c\):
\[
a(b + c) = ab + ac
\]
In this case:
- \(a = -8\)
- \(b = -x^2\)
- \(c = 7\)
So, Emily distributed \(-8\) across \(-x^2 + 7\):
\[
-8(-x^2) + (-8)(7) = 8x^2 - 56
\]
However, note that Emily's initial calculation must be focusing on simplifying that alongside the addition of \(6.\)
Instead of computing it directly, when combined with the rest of the equation, the expression would ultimately yield to the addition of \(6\) which will lead to finding the constant terms.
Either way, the justification for the first step (before further simplifications) lies within the application of the Distributive Property.
The original equation is:
\[
-8(-x^2 + 7) + 6 = -5x^2 + 4
\]
In the first step, Emily simplified the left side of the equation:
\[
-8(-x^2 + 7) = -5x^2 - 2
\]
Here, Emily applied the **Distributive Property**. The Distributive Property states that for any numbers \(a\), \(b\), and \(c\):
\[
a(b + c) = ab + ac
\]
In this case:
- \(a = -8\)
- \(b = -x^2\)
- \(c = 7\)
So, Emily distributed \(-8\) across \(-x^2 + 7\):
\[
-8(-x^2) + (-8)(7) = 8x^2 - 56
\]
However, note that Emily's initial calculation must be focusing on simplifying that alongside the addition of \(6.\)
Instead of computing it directly, when combined with the rest of the equation, the expression would ultimately yield to the addition of \(6\) which will lead to finding the constant terms.
Either way, the justification for the first step (before further simplifications) lies within the application of the Distributive Property.