To understand which property justifies Anne's first step, let's carefully analyze the original equation and the first step she took.
Original Equation:
[tex]\[ 7x^2 + (8x^2 + 1) = 6x^2 - 5 \][/tex]
First Step:
[tex]\[ 15x^2 + 1 = 6x^2 - 5 \][/tex]
Anne combined the terms on the left side of the equation. Let's see this more closely:
1. She had [tex]\(7x^2\)[/tex] and [tex]\((8x^2 + 1)\)[/tex].
2. By combining [tex]\(7x^2\)[/tex] and [tex]\(8x^2\)[/tex], she got [tex]\(15x^2\)[/tex].
3. The constant [tex]\(1\)[/tex] remains unchanged in the equation.
The key here is combining the like terms, [tex]\(7x^2\)[/tex] and [tex]\(8x^2\)[/tex], into a single term, [tex]\(15x^2\)[/tex].
The property used to combine these like terms is the associative property of addition. This property states that the way in which numbers are grouped in an addition problem does not change the sum.
Thus, grouping the terms [tex]\(7x^2\)[/tex] and [tex]\(8x^2\)[/tex] together inside the parentheses and combining them to form [tex]\(15x^2\)[/tex] is justified by the associative property of addition.
Therefore, the correct answer is:
associative property of addition
