Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-20,\) solve each equation and check. $$ 0.4 x+8=0.5 x $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 80 \).

Step 1: Isolate Variable Terms

First, we want to get all terms involving the variable on one side of the equation and constant terms on the other. Start by subtracting \(0.4x\) from both sides of the equation:\[0.4x + 8 - 0.4x = 0.5x - 0.4x\]This simplifies to:\[8 = 0.1x\]

Step 2: Solve for the Variable

Now that we have \(0.1x\) isolated, divide both sides by 0.1 to solve for \(x\):\[\frac{8}{0.1} = x\]Thus, \(x = 80\).

Step 3: Verify the Solution

Substitute \(x = 80\) back into the original equation to check the solution:\[0.4(80) + 8 = 0.5(80)\]Calculate each side:\[32 + 8 = 40 \,\, \text{and} \,\, 0.5 imes 80 = 40\]Both sides are equal, confirming that \(x = 80\) is the correct solution.

Key Concepts

Isolating Variable Terms

When solving linear equations, the first step is to have all terms containing the variable on one side of the equation. This makes it easier to work with the equation and find the value of the variable. In the problem \(0.4x + 8 = 0.5x\), the variable \(x\) appears on both sides of the equation. Our goal is to "isolate" the terms with \(x\).To do this, we subtract \(0.4x\) from both sides of the equation. This results in:

• Left side: \(0.4x + 8 - 0.4x = 8\)
• Right side: \(0.5x - 0.4x = 0.1x\)

Thus, the equation simplifies to \(8 = 0.1x\). Now, the variable terms are isolated, and the next step can be taken to solve this simplified equation.

Solving for Variables

With the variable term isolated on one side of the equation, the next step is figuring out the value of the variable. We have the equation \(8 = 0.1x\). To solve for \(x\), we need to perform operations to make \(x\) stand alone. This involves dividing both sides of the equation by \(0.1\) to "solve" for \(x\):\[\frac{8}{0.1} = x\]Calculating this, we find that \(x = 80\). This step effectively turns the complex equation into a simple statement of value, making it clear what \(x\) represents. Each operations step reduces the complexity, helping to keep track of mathematical manipulations.

Equation Verification

After solving for the variable, the final step is to verify the solution. This ensures that our calculations are correct, and the value found for \(x\) satisfies the original equation. Let's substitute \(x = 80\) back into the original equation:\[0.4(80) + 8 = 0.5(80)\]Calculating separately:

• \(0.4 \times 80 = 32\)
• Adding \(32 + 8 = 40\)
• On the other side, \(0.5 \times 80 = 40\)

Both sides equal 40, confirming the equation holds true with \(x = 80\). This step is crucial as it acts as a check against any possible errors made during problem-solving, ensuring reliability in the solution.