olve and justify your answer using the properties of operations (i.e Distributive property, associative property, commutative property, identity property, substitution, ect…)
-6y + 8 = y +9 - 7y - 1
To solve the equation -6y + 8 = y + 9 - 7y - 1, we can start by combining like terms on both sides of the equation.
On the right side, we have y and -7y, which are like terms. So, we can combine them to get y - 7y = -6y.
Now, the equation becomes -6y + 8 = -6y + 9 - 1.
Next, we can combine the constants 9 and -1 to get 9 - 1 = 8.
The equation now becomes -6y + 8 = -6y + 8.
At this point, we can see that both sides of the equation are equal. This means that the equation has infinitely many solutions.
To justify this answer using the properties of operations:
1. Commutative Property: The commutative property of addition allows us to change the order of the terms. In this equation, we switched the order of y and -7y, which did not change their sum. This property is used when combining like terms.
2. Associative Property: The associative property of addition allows us to regroup terms in an equation. In this equation, we used this property to regroup the terms -6y and 8 on both sides of the equation. This property is used when combining like terms.
3. Identity Property: The identity property of addition states that the sum of any number and 0 is that number itself. In this equation, we do not directly use the identity property, as there are no instances of adding or subtracting 0. However, the equation shows that the left side is equal to the right side, indicating that the equation holds true.
4. Distributive Property: The distributive property allows us to multiply a number by each term in a group. In this equation, we do not use the distributive property.
5. Substitution: We did not use substitution in solving this equation. However, substitution can be used to check the solution if one is found.
By combining like terms and simplifying the equation, we obtained the equation -6y + 8 = -6y + 8. This equation shows that both sides are equal, indicating that the equation has infinite solutions.