Problem 64 - Algebra 2

Question

Solve each equation. \(7 w+2=3 w-6\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Equation

We are given the equation to solve: \(7w + 2 = 3w - 6\). Our goal is to find the value of \(w\) that makes this equation true.

Step 2: Move the Variable Terms

Subtract \(3w\) from both sides of the equation to get all the \(w\) terms on one side. This gives us: \[ 7w - 3w + 2 = -6 \] or simplified, \[ 4w + 2 = -6 \].

Step 3: Isolate the Variable

Subtract 2 from both sides to isolate the \(w\) term: \[ 4w + 2 - 2 = -6 - 2 \] which simplifies to \[ 4w = -8 \].

Step 4: Solve for the Variable

Divide both sides of the equation by 4 to solve for \(w\): \[ w = \frac{-8}{4} \], which simplifies to \[ w = -2 \].

Key Concepts

Solving Equations

Solving equations is like solving a puzzle where we need to find the value of an unknown variable that makes the entire equation true. In the equation \(7w + 2 = 3w - 6\), our unknown variable is \(w\). The main objective is to unravel this puzzle step by step.

Each equation can have multiple steps, but they are just a series of operations to systematically work our way towards the solution. We aim to perform all operations without changing the equality, ensuring the left side of the equation equals the right side. Whether it’s addition, subtraction, multiplication, or division, every operation should steadily push us closer to isolating the variable.

Isolating Variables

When we talk about isolating variables, we mean rearranging the equation so that the unknown variable stands alone on one side of the equation. This is crucial because it allows us to see directly what the variable equals.

In our example, after moving terms around, we get \(4w + 2 = -6\). The aim here is to make sure that \(w\) is sitting pretty by itself on the left side of the equation. To do this, we need to get rid of the number \(2\) that’s added to \(4w\).

Steps to Isolate

• First, subtract \(2\) from both sides: \(4w + 2 - 2 = -6 - 2\).
• As a result, we get \(4w = -8\).

Now, the equation is prepared for the final step of solving for \(w\).

Algebraic Manipulation

Algebraic manipulation is a fundamental tool in solving equations, and it involves using algebraic properties and operations to change the form of an equation to make it easier to solve. This process can include combining like terms, using distributive properties, and more.

In our equation, we used algebraic manipulation to simplify terms and get all \(w\) variables together. We started with the rearrangement \(7w - 3w + 2 = -6\), simplifying this to \(4w + 2 = -6\).

Final Steps

Once the equation is simplified, algebraic manipulation also involves dividing or multiplying both sides of the equation to solve for the variable. Here, dividing both sides by \(4\) in the equation \(4w = -8\), gives us \(w = -2\).

This manipulation helps transform complex equations into easier, solvable ones.