Final answer:
Yes, the equation 5 + 5 + m = 10 + m remains balanced after adding 'm' to both sides due to the Addition Property of Equality, which keeps the equation equal when the same value is added to both sides.
Step-by-step explanation:
The question posed by the student involves checking whether an algebraic equation remains balanced after adding a variable to both sides. Initially, Noah wrote the equation 5 + 5 = 10, which is a balanced equation since both sides equal 10. Then he wrote the equation 5 + 5 + m = 10 + m. To determine if this equation is still balanced, we need to understand the properties of equality.
In mathematics, one such property is the Addition Property of Equality, which states that if you add the same number (or variable) to both sides of an equation, the equation remains balanced. This means that whatever value 'm' might represent, adding it to both sides does not change the equality of the equation. When you perform the same operation on both sides of an equation, you are maintaining the balance, just like in a physical balance scale.
To link this to the student's examples of balancing chemical equations, writing balanced reactions is essential to respect the law of conservation of matter. On a chemical scale, atoms are neither created nor destroyed, so the same number of atoms must be present on both sides of the equation. Likewise, with numbers, performing the same mathematical operation (such as adding 'm') to both sides of an equation preserves the equality.
Thus, when Noah added 'm' to both sides of the original equation, the new equation 5 + 5 + m = 10 + m indeed remained balanced. No matter what value 'm' represents, the two sides of the equation will have the same value, given that 'm' is added to both sides equally.
To summarize, equations remain balanced as long as the same operations are applied to both sides, reflective of fundamental mathematical principles that apply regardless of cultural or temporal differences. Therefore, the answer is Yes, the equation is still balanced after adding 'm' to both sides.