Problem 2 - Algebra for College Students

Question

Solve each equation. \(1.6 x=8\)

Step-by-Step Solution

Answer

\(x = 5\)

Step 1: Understand the Equation

The given equation is \(1.6x = 8\). This equation shows that \(x\) is multiplied by 1.6 to get 8. Our goal is to find the value of \(x\).

Step 2: Isolate the Variable

To find \(x\), we need to eliminate the 1.6 that is multiplying it. We can do this by dividing both sides of the equation by 1.6.

Step 3: Perform the Division

Divide each side of the equation by 1.6:\[ x = \frac{8}{1.6} \] Simplifying this, we get: \[ x = 5 \]

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(x = 5\) back into the original equation: \[ 1.6 \times 5 = 8 \] This statement holds true, confirming that \(x = 5\) is indeed the correct solution.

Key Concepts

Equation Isolation

When solving a linear equation, one of the first steps is to isolate the variable. In our example with the equation \(1.6x = 8\), the aim is to rearrange the equation so \(x\) stands alone on one side. This allows us to clearly see the solution for \(x\). To isolate the variable, you need to "undo" the operation that is being performed on it. Here, \(x\) is being multiplied by 1.6. Thus, our task is to reverse this multiplication operation.

• Identify the operation: Multiplication by 1.6
• Reverse this operation: Use division to counteract multiplication

By understanding the process of isolation, you can manipulate equations to find solutions effectively.

Division in Equations

Division is a crucial operation when solving equations that involve multiplication of the variable by a constant. Once we have identified the need to undo multiplication in the equation \(1.6x = 8\), division serves as our tool. We achieve this by dividing both sides of the equation by the same number. This maintains the equality of the equation while isolating \(x\):\[ x = \frac{8}{1.6} \]This division gives us:

• Understanding division as reversing multiplication ensures balance
• Performing careful calculation: \(8/1.6 = 5\)

This step requires precision as it leads directly to our solution.

Verification of Solutions

Verifying solutions is an essential step in solving equations to ensure that the answer obtained is indeed correct. After we found \(x = 5\), it’s important to confirm that this value satisfies the original equation.To verify, substitute \(x = 5\) back into the initial equation:\[ 1.6 \times 5 = 8 \]Checking involves:

• Replacing the variable with the solution
• Ensuring the equation simplifies to a true statement: \(8 = 8\)

By verification, we gain confidence in our calculations and conclusion, confirming that \(x = 5\) is indeed the correct and valid solution.