Problem 26 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ \frac{-w}{5}=1 $$

Step-by-Step Solution

Answer

The solution is \( w = -5 \).

Step 1: Isolate the Variable

To solve the equation \( \frac{-w}{5} = 1 \), we begin by isolating the variable \( w \). To do this, multiply both sides of the equation by 5 to eliminate the denominator, which gives us:\[ -w = 5 \]

Step 2: Solve for the Variable

Next, solve for \( w \) by multiplying both sides by -1 to get rid of the negative sign in front of \( w \). This results in:\[ w = -5 \]

Step 3: Verify the Solution

Finally, substitute \( w = -5 \) back into the original equation to verify the solution:\[ \frac{-(-5)}{5} = \frac{5}{5} = 1 \]Since both sides of the equation are equal, the solution \( w = -5 \) is verified as correct.

Key Concepts

Isolate the Variable

When solving linear equations, the first crucial step is to isolate the variable. This simply means to get the variable, often denoted as "w", "x", "y", etc., all by itself on one side of the equation. Let's illustrate this using the equation from our problem: - Start with: \( \frac{-w}{5} = 1 \)- To isolate "w", first eliminate the fraction by multiplying both sides by 5. This means performing the same operation on both sides to maintain equality. The whole idea is to simplify the equation step-by-step:- Result: \( -w = 5 \)Now "w" is nearly isolated, but it still has a negative sign. To fully isolate "w", multiply both sides by -1: - Result: \( w = -5 \)Isolating the variable effectively uncovers the value of the unknown in the equation. It requires understanding operations like addition, subtraction, multiplication, and division, and using them strategically until the variable stands alone.

Equation Verification

Once we've isolated and solved for the variable, it's important to verify that our solution is correct. Verification ensures that no mistakes were made during the process. This is done by substituting the solution back into the original equation.- Start with the original equation: \( \frac{-w}{5} = 1 \)- Substitute the solution, \( w = -5 \), back into the equation:- Calculating: \( \frac{-(-5)}{5} = \frac{5}{5} \)- Simplify: \( 1 = 1 \)Since both sides of the equation match, the solution is confirmed as correct. Verification has two important roles:

• It minimizes errors by checking the accuracy of your solution.
• It reinforces understanding by retracing the logical steps used to find the solution.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill for solving equations. It involves reducing equations to their simplest form while maintaining the same value, making it easier to work with them.In our exercise:- We began with the equation: \( \frac{-w}{5} = 1 \).- By multiplying both sides by 5, we simplified the fraction and got:- Simplified: \( -w = 5 \).Further simplifying involved turning \( -w = 5 \) into \( w = -5 \) by getting rid of the negative sign with multiplication by -1.Key aspects of simplifying include:

• Clearing fractions by multiplying by the denominator.
• Eliminating negative signs where necessary to make the variable positive.
• Combining like terms and reducing operations to reveal the variable.

Mastering these steps makes solving equations smoother and more intuitive.