91Ó°ÊÓ

To start, let's understand what linear equations are. Linear equations are mathematical statements where the highest exponent of the variable (often represented by letters like \(x\) or \(y\)) is one. These equations appear as straight-line graphs on the coordinate plane.
In this problem, we set up linear equations to represent Samantha's salary options. We have two options: \(\(1800\) fixed salary and \(\)1600 + 0.04x\) where \(x\) is the total sales.
Both salary options are linear because the variable \(x\) is raised to the power of one. We aim to find the value of \(x\) where both options give the same total pay for Samantha, making the linear equations equal.

Next, let's discuss solving for the variable, which in our case is \(x\). Solving for a variable means finding its value that makes the equation true.
Here's the process step-by-step:

• We start by setting the two salary equations equal: \(1800 = 1600 + 0.04x\).
• To isolate \(x\), subtract 1600 from both sides: \(1800 - 1600 = 0.04x\).
• This simplifies to \(200 = 0.04x\).
• Finally, we divide both sides by \(0.04\) to solve for \(x\), so \(x = \frac{200}{0.04} = 5000\).

This process shows how to isolate and solve for \(x\), revealing that Samantha needs to sell \($5000\) worth of furniture for both salary options to equal.

Let's look at the concept of commissions. A commission is a payment based on the amount of sales made, usually a percentage of the total sales.
In this scenario, Samantha has an option between a purely fixed salary or a combination of base salary and commission.
Her commission rate is \(4\%\), or \(0.04\) in decimal form. This rate means for every dollar of furniture sold, Samantha earns an additional \(4\) cents.
So, if Samantha sells \(\$5000\) worth of furniture, she would earn:\[\$1600 + 0.04 \times 5000\ = \$1800\]. This means the two salary options would be equal if she sells \(\$5000\) worth of furniture. Understanding commissions helps in evaluating such payment plans effectively.