Problem 122 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 5 w-6=4+2 w $$

Step-by-Step Solution

Answer

The solution is \( w = \frac{10}{3} \).

Step 1: Isolate the variable terms

First, move variable terms to one side. Start by subtracting \(2w\) from both sides of the equation to isolate the \(w\) terms on the left: \( 5w - 2w - 6 = 4 \). This simplifies to \( 3w - 6 = 4 \).

Step 2: Remove constant terms

Next, isolate the \(w\) term by eliminating the constant on its side. Add 6 to both sides of the equation to remove the \(-6\): \( 3w - 6 + 6 = 4 + 6 \). This simplifies to \( 3w = 10 \).

Step 3: Solve for the variable

Finally, solve for \(w\) by dividing both sides of the equation by 3 to get:\( w = \frac{10}{3} \).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra, and it's like solving a puzzle where you need to find the missing piece. The goal is to determine which value of the variable makes the equation true. When you encounter an equation like the example provided, it's crucial to follow a series of systematic steps.

• Begin by organizing the equation: align similar terms on each side of the equation.
• Simplify the equation step by step, ensuring you perform the same operation to both sides to maintain balance.
• Continue simplifying until you isolate the variable and solve for it, revealing the solution.

Remember, practice makes perfect. The more you work with equations, the more intuitive solving them becomes.

Isolating Variables

Isolating variables is a critical step in solving equations. This step involves manipulating the equation to get the variable alone on one side, effectively uncovering the mystery number. To isolate a variable:

• Combine like terms to get all variable terms on one side and constants on the other.
• Use arithmetic operations strategically, such as addition or subtraction, to move terms from one side to the other.
• The aim is to simplify the equation step by step, eventually getting the variable by itself.

In our specific example, the variable term was isolated by subtracting equal parts from each side, and then adding to counteract constants. This systematic approach will help reveal the unknown variable without error.

Linear Equations

Linear equations are equations that graph as straight lines. They are often in the form of \[ ax + b = c \] where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Here are some key properties:

• The solutions to these equations are straightforward, often found by simple arithmetic manipulation.
• Linear equations have one unique solution.
• Solving them usually involves straightforward steps such as combining like terms and isolating the variable.

Understanding linear equations is essential as they form the basis for more complex algebraic expressions. Mastery of linear equations will greatly benefit more advanced algebra studies.