Problem 109 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{w}{2}-15=4 $$

Step-by-Step Solution

Answer

The solution is \(w = 38\).

Step 1: Isolate the Fraction Term

To solve \(\frac{w}{2}-15=4\), we first need to isolate the term with the fraction. We can do this by adding 15 to both sides of the equation. \( \frac{w}{2} = 4 + 15 \) Now simplify the right side: \( \frac{w}{2} = 19 \)

Step 2: Eliminate the Fractional Coefficient

Now that \( \frac{w}{2} = 19 \), we eliminate the fraction by multiplying both sides by 2, which is the denominator of the fraction. \( w = 19 \times 2 \) This simplifies to:\( w = 38 \)

Step 3: Verify the Solution

To check our work, substitute \(w = 38\) back into the original equation: \( \frac{38}{2} - 15 = 4 \)Calculate the left side: \( 19 - 15 = 4 \)Since both sides are equal, \(w = 38\) is indeed the correct solution.

Key Concepts

Equation Isolation

When solving linear equations, isolating the variable is often the first crucial step. This means we want to "free" the variable from other numbers and terms so we can solve for it easily. In the problem \(\frac{w}{2} - 15 = 4\), our goal is to isolate the fraction term \(\frac{w}{2}\). To do this, we add 15 to both sides of the equation. This action removes the -15 that is subtracted from \(\frac{w}{2}\), essentially getting us closer to having the variable alone on one side. By adding or subtracting terms on both sides of the equation, you maintain the balance, much like ensuring both sides of a scale stay even. Here's the step illustrated:

• Original equation: \(\frac{w}{2} - 15 = 4\)
• Add 15 to both sides: \(\frac{w}{2} = 4 + 15\)
• Simplified result: \(\frac{w}{2} = 19\)

By isolating \(\frac{w}{2}\), we simplify the problem, making it easier to solve for \(w\), the variable in question.

Fractional Coefficients

Fractional coefficients can be tricky, but they're manageable with the right techniques. A fractional coefficient means that the variable is being divided by a number, in our exercise, it is \(\frac{w}{2}\). The goal is to eliminate the fraction to find the value of the variable, \(w\). This is achieved by performing the opposite operation of what's being done in the fraction. Here, since \(w\) is divided by 2, we multiply both sides of the equation by 2 to counteract the division:

• Starting equation: \(\frac{w}{2} = 19\)
• Multiply both sides by 2: \(w = 19 \times 2\)
• Result: \(w = 38\)

By removing the fractional coefficient, \(w\) is isolated as a whole number, simplifying the equation. This technique of multiplying to clear a fraction makes solving equations much more straightforward.

Equation Verification

Verifying your solution is a key part of solving an equation. Verification ensures your solution is correct and satisfies the original problem. For our exercise, once we calculated \(w = 38\), we need to check if this value, when substituted back into the original equation, fulfills the equation:

• Original equation: \(\frac{w}{2} - 15 = 4\)
• Substitute \(w = 38\): \(\frac{38}{2} - 15 = 4\)
• Calculate: \(19 - 15 = 4\)

Since both sides of the equation are equal after substitution, the solution \(w = 38\) is confirmed as correct. This process of verification prevents errors and ensures confidence in the correctness of your solution.