Problem 32 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$-9 w+6=0$$

Step-by-Step Solution

Answer

The solution to the equation \(-9w+6=0\) is \(w = 2/3\).

Step 1: Solve for w

Isolate w on one side of the equation. \(-9w + 6 = 0\) can be rewritten as \(-9w = -6\) by subtracting 6 from both sides of the equation. Then divide both sides of the equation by -9 to solve for w. Thus, \(w = -6 / -9 = 2/3\).

Step 2: Check the solution

Substitute \(w = 2/3\) back into the original equation to verify if it is a solution. The left-hand side of the original equation is \(-9w + 6\), which becomes \(-9*(2/3) + 6 = -6 + 6 = 0\). Since the left-hand side equals the right-hand side, the solution is correct.

Key Concepts

Checking Solutions

When you solve a linear equation, it's crucial to check your solution. This ensures that your answer is accurate and satisfies the original equation. After you've found a value for the variable, substitute it back into the original equation to see if both sides of the equation are equal.

• Substitute the solution back into the left-hand side of the equation.
• Calculate each part step by step to avoid errors.
• Compare it with the right-hand side.

If both sides are equal, your solution is correct. In our example, when we plug in \(w = \frac{2}{3}\), it satisfies the equation. Make this a habit to ensure accuracy in all your math work.

Isolation of Variables

The process of isolation in solving equations involves rearranging the equation to get the variable alone on one side. This is primarily done through arithmetic operations that maintain the balance of the equation. In the given exercise, we isolate \(w\) by performing two main actions:

• Subtract 6 from both sides to remove the constant term.
• Divide by \(-9\) to eliminate the coefficient of \(w\).

These steps simplify the equation to \(w = \frac{2}{3}\). Clear understanding and application of these isolation steps are essential for solving any equation effectively.

Equation Verification

Verification is the step where you confirm the solution's validity by substituting it back into the original equation. This step is not just double-checking but also serves as proof that the value found for the variable truly satisfies the equation.
Substitute your solution back into both sides, and compute the values. If both sides match, this verifies your solution is correct.
In our example, the solution \(w = \frac{2}{3}\) was put back into the original equation \(-9w + 6 = 0\) and simplified to get \(0 = 0\). This verification shows the solution we found is indeed correct!

Fraction Arithmetic

Handling fractions can be tricky, yet they're a vital part of solving many equations. Fraction arithmetic involves adding, subtracting, multiplying, and dividing fractions. In our equation, once we isolated \(w\), the subsequent division resulted in a fraction, \(\frac{2}{3}\).
When you divide by a negative number, the signs cancel out, ensuring the fraction is simplified correctly.

• Always simplify fractions to their lowest terms.
• Be mindful of the signs when multiplying or dividing.
• Ensure your final result makes sense in the context of the problem.

Mastering fraction arithmetic is fundamental for success in solving equations, as fractions often appear in solutions and during intermediate steps.