Absolutely, let's look at the process Bianca followed in solving the equation.
Starting with the original equation:
[tex]\[ 6x^2 + \left(3x^2 + 9\right) = -10x^2 + 10 \][/tex]
To get to the first step:
[tex]\[ 9x^2 + 9 = -10x^2 + 10 \][/tex]
Bianca has combined the terms [tex]\(6x^2\)[/tex] and [tex]\(3x^2\)[/tex] on the left side of the equation. This combination is valid because it utilizes a fundamental property in solving equations, which is:
The Addition Property of Equality.
This property states that you can add the same number (or expression) to both sides of an equation without changing the equality. The key is to keep the equation balanced. By combining like terms on the left-hand side:
1. Combine [tex]\(6x^2\)[/tex] and [tex]\(3x^2\)[/tex] to get [tex]\(9x^2\)[/tex].
2. The remaining term ([tex]\(+9\)[/tex]) is already simplified.
So the equation [tex]\(6x^2 + 3x^2 + 9\)[/tex] simplifies to [tex]\(9x^2 + 9\)[/tex].
Thus, we reach the next step in the equation:
[tex]\[ 9x^2 + 9 = -10x^2 + 10 \][/tex]
In conclusion, the property justifying Bianca’s first step is the addition property of equality.
