Problem 19 - Prealgebra

Question

Solve equation. Check your solution. \(3.1 w+5=0.8+w\)

Step-by-Step Solution

Answer

The solution is \( w = -2 \).

Step 1: Isolating the variable on one side

First, we need to get all the terms containing the variable on one side. We have the equation \( 3.1w + 5 = 0.8 + w \). To isolate \( w \), subtract \( w \) from both sides of the equation: \( 3.1w + 5 - w = 0.8 + w - w \). This simplifies to \( 2.1w + 5 = 0.8 \).

Step 2: Solving for the variable

Next, we solve for \( w \) by isolating it completely. From \( 2.1w + 5 = 0.8 \), subtract 5 from both sides to move the constant term: \( 2.1w + 5 - 5 = 0.8 - 5 \). This simplifies to \( 2.1w = -4.2 \). Now, divide both sides by 2.1 to solve for \( w \): \( w = \frac{-4.2}{2.1} \). This simplifies to \( w = -2 \).

Step 3: Checking the solution

To verify the solution, substitute \( w = -2 \) back into the original equation. The original equation is \( 3.1w + 5 = 0.8 + w \). Substituting \( w = -2 \), we have \( 3.1(-2) + 5 = 0.8 + (-2) \). This simplifies to \( -6.2 + 5 = 0.8 - 2 \). Both sides simplify further to \( -1.2 = -1.2 \), confirming our solution is correct.

Key Concepts

Solving Equations

Solving linear equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. The goal is to isolate the variable on one side of the equation. This often requires moving terms around by performing the same operation on both sides. For example, if our equation is \(3.1w + 5 = 0.8 + w\), we start by ensuring all terms with the variable are on one side.

A common approach is to subtract or add terms to both sides. In this case, subtracting \(w\) from both sides helps isolate the variable better, simplifying our equation to \(2.1w + 5 = 0.8\). This step is crucial as it breaks down the equation into simpler parts, leading us to find the value of \(w\). Once terms are simplified, other basic operations, like multiplication or division, are typically used to solve for the variable.

• Subtract: Remove terms from one side to balance the equation
• Add: Compensate for subtracted terms to maintain equality
• Multiply/Divide: Simplify variable terms to isolate the variable

This systematic breakdown leads us to the solution \(w = -2\).

Equation Verification

After solving an equation, it's crucial to verify the solution to ensure it's correct. Verification involves substituting the found value back into the original equation. This helps confirm whether the left-hand side (LHS) equals the right-hand side (RHS) of the equation.

Consider our example where \(w = -2\). We substitute \(-2\) back into the original equation: \(3.1(-2) + 5\) on the LHS and \(0.8 + (-2)\) on the RHS. Simplifying both sides, we find each resolves to \(-1.2\), proving that our solution holds true.

• Substitute: Replace the variable with your found solution
• Simplify: Calculate both sides to check equality
• Verify: Confirm both sides equal, thereby affirming the solution

Verification ensures that the steps you've followed in solving the equation are accurate and error-free.

Algebraic Manipulation

Algebraic manipulation involves altering an equation to make it easier to solve. This manipulation requires a strong understanding of algebraic rules and properties, such as distribution, combining like terms, and the inverse operations of addition and multiplication.

In solving \(3.1w + 5 = 0.8 + w\), we used algebraic manipulation to both isolate \(w\) and combine like terms. By subtracting \(w\) from both sides, we reduced complex terms into simpler ones: \(2.1w\). This manipulation places the equation into a more straightforward format, where basic arithmetic like division quickly leads to the solution. Without manipulation, equations remain too complex to solve efficiently.

• Isolate: Move terms to isolate the variable easily
• Combine: Add or subtract like terms to simplify
• Apply properties: Use distributive or inverse operations as needed

These strategies create a clear path to resolve equations, establishing a logical flow from complex to simple expressions.