Watch this animation to see the answer.
Why does the elimination method work?
The next step in solving the system is to add the equations together.
$\begin{array}{r}
12 x+3 y=14 \\
+6 x-3 y=69 \\
\hline
\end{array}$
This works because of the
A. multiplication property of equality
B. commutative property
C. addition property of equality
D. zero identity
Answer (1)
The problem involves adding two equations together in the elimination method.
The addition property of equality justifies this step.
The addition property of equality states that if a = b and c = d , then a + c = b + d .
Therefore, the correct answer is C. addition property of equality. C
Explanation
Understanding the Problem We are given two equations, 12 x + 3 y = 14 and 6 x − 3 y = 69 , and we are adding them together. The question asks us to identify the property that justifies this step.
Recalling the Addition Property of Equality The addition property of equality states that if a = b and c = d , then a + c = b + d . In other words, we can add equal quantities to both sides of an equation, and the equality will still hold.
Applying the Property to the Equations In our case, we have 12 x + 3 y = 14 and 6 x − 3 y = 69 . We are adding the left-hand sides ( 12 x + 3 y and 6 x − 3 y ) and the right-hand sides ( 14 and 69 ). This is a direct application of the addition property of equality.
Conclusion Therefore, the correct answer is C. addition property of equality.
Examples
The addition property of equality is a fundamental concept in algebra. It's used in balancing equations, such as in chemistry when balancing chemical equations or in physics when analyzing forces. For example, if you have a balanced scale with weights on each side, adding the same weight to both sides keeps the scale balanced. This principle is also used in economics to analyze supply and demand curves, where changes on one side of the equation must be balanced by changes on the other side to maintain equilibrium.
