Question

to begin solving this system of linear equations by elimination, you can add the equations.
$\

$$\begin{align} 3x + 2y &= 38 \\\\ + \\ 5x - 2y &= -30 \\\\ \\hline 8x &= 8 \\end{align}$$

$
this step works because of the _____.
\bigcirc a. addition property of equality
\bigcirc b. multiplication property of equality
\bigcirc c. distributive property
\bigcirc d. commutative property

Explanation:

To determine the property used, we analyze each option:

• Option A: The addition property of equality states that if \(a = b\) and \(c = d\), then \(a + c = b + d\). Here, we have two equations \(3x + 2y = 38\) (so \(a = 3x + 2y\), \(b = 38\)) and \(5x - 2y = - 30\) (so \(c = 5x - 2y\), \(d=-30\)). Adding them gives \(8x = 8\), which follows the addition property of equality.
• Option B: The multiplication property of equality involves multiplying both sides of an equation by a non - zero number, which is not what is happening here.
• Option C: The distributive property is about \(a(b + c)=ab+ac\), and it is not relevant to adding two equations.
• Option D: The commutative property is about the order of addition or multiplication (e.g., \(a + b=b + a\) or \(ab = ba\)), and it does not apply to adding two equations.

Answer:

A. addition property of equality