Question

which is the best first step when solving the following system of equations?

\\(y = 7x + 3\\)
\\(y = 9x\\)

• multiply the first equation by \\(-7\\).
• substitute \\(9x\\) for \\(y\\) in the first equation.
• multiply the second equation by \\(-3\\).
• substitute \\(7x\\) for \\(y\\) in the second equation.

Explanation:

Analyze the given system

We are given a System of Equations:

$$
LATEXBLOCK0
$$

Our goal is to find the best first step to solve this system.

Evaluate the substitution method

Notice that the second equation explicitly defines \(y\) in terms of \(x\):

$$ y = 9x $$

Since \(y\) is already isolated, we can use the Substitution Method by replacing \(y\) in the first equation with the expression \(9x\).

Compare with the given options

Let's check the options:

• Multiply the first equation by \(-7\): This does not help isolate variables easily.
• Substitute \(9x\) for \(y\) in the first equation: This directly replaces \(y\) with \(9x\), giving \(9x = 7x + 3\), which is a simple single-variable equation. This is the best first step.
• Multiply the second equation by \(-3\): This does not simplify the system.
• Substitute \(7x\) for \(y\) in the second equation: This is incorrect because \(y\) is equal to \(7x + 3\), not just \(7x\).

Answer:

• Multiply the first equation by \(-7\).
• Substitute \(9x\) for \(y\) in the first equation. (Correct answer)
• Multiply the second equation by \(-3\).
• Substitute \(7x\) for \(y\) in the second equation.