Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution in. \(0.6(x+300)=0.65 x-205\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 7700\)

Step 1: Expand and Simplify

By expanding the left-hand side of the equation, we have \(0.6 \cdot x + 0.6 \cdot 300 = 0.65x - 205\). This simplifies to \(0.6x + 180 = 0.65x - 205\)

Step 2: Isolate x terms

Subtract \(0.6x\) from both sides and add \(205\) to both sides. The equation now becomes \(0.05x = 385\)

Step 3: Solve for x

To find the value of x, we divide both sides of the equation by 0.05. The solution for x is approximately \(x = 7700\).

Step 4: Verify Solution

Substitute the solution back into the original equation. Doing this yields \(0.6(7700 + 300) = 0.65 \cdot 7700 - 205\), which results in a valid equation (4800 = 4800), confirming that our solution is correct.

Key Concepts

Expanding and Simplifying Equations

When you first encounter an algebraic equation like \(0.6(x+300)=0.65 x-205\), it might look a bit overwhelming. But don't worry! The first step is to expand and simplify it. Expansion means distributing any numbers outside parentheses across the terms inside.

In this case, you have \(0.6(x + 300)\). To expand, multiply each term inside the parentheses by \(0.6\). This gives \(0.6 \cdot x + 0.6 \cdot 300 = 0.65x - 205\).

Simplifying means combining like terms and making the equation easier to read. Here, the left side simplifies to \(0.6x + 180\). Now your equation looks much clearer: \(0.6x + 180 = 0.65x - 205\). After this, you are fully set to move to the next step of solving the equation.

Isolating Variables

The goal of isolating variables is to have the variable on one side of the equation and numbers on the other. This allows you to see exactly what the unknown variable equals.

Start with the equation you've expanded: \(0.6x + 180 = 0.65x - 205\). To isolate \(x\), begin by getting all \(x\) terms on one side. Here, subtract \(0.6x\) from both sides, giving \(180 = 0.05x - 205\).

Next, you want to eliminate the constant term (\(-205\)) on the right side by adding \(205\) to both sides. This results in \(385 = 0.05x\).

With \(0.05x = 385\), you now have a much simpler equation where \(x\) can be easily solved. Divide both sides by \(0.05\) to isolate \(x\). This operation gives you \(x = 7700\), the solution to the equation.

Verification of Solution

Verification is a crucial step to ensure that your solution is correct. It involves substituting your solution back into the original equation to see if it satisfies it.

So, let's verify the solution \(x = 7700\). Substitute \(7700\) back into the original equation: \(0.6(7700 + 300) = 0.65 \cdot 7700 - 205\). Calculate each side separately:

• Left side: \(0.6 \cdot 8000 = 4800\)
• Right side: \(0.65 \cdot 7700 - 205 = 4800\)

Both sides are equal, \(4800 = 4800\), which confirms that the solution is indeed correct.

Never skip the verification step! It's the best way to make sure you haven't made any errors during the calculation process.