Problem 15 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(\frac{1}{2} x-8=1\)

Step-by-Step Solution

Answer

The solution is \(x = 18\).

Step 1: Isolate the Variable Term

First, isolate the term containing the variable on one side of the equation. In this case, add 8 to both sides of the equation to get rid of the constant term on the left:\[ \frac{1}{2}x - 8 + 8 = 1 + 8 \]This simplifies to:\[ \frac{1}{2}x = 9 \]

Step 2: Solve for the Variable

Now, solve for the variable \(x\). To do this, we need to eliminate the fraction by multiplying both sides of the equation by 2:\[ 2 \times \frac{1}{2}x = 2 \times 9 \]This gives us:\[ x = 18 \]

Step 3: Verify the Solution

Substitute \(x = 18\) back into the original equation to check our solution:\[ \frac{1}{2}(18) - 8 = 1 \]Calculate \(\frac{1}{2}(18)\):\[ 9 - 8 = 1 \]This confirms that the left-hand side equals the right-hand side, so \(x = 18\) is the correct solution.

Key Concepts

Solving Equations

Solving equations is all about finding the value of the variable that makes the equation true. For instance, in the equation \( \frac{1}{2}x - 8 = 1 \), our job is to determine the value of \( x \) that satisfies this equation.When solving equations, think of it as balancing a scale. Each side of the equation should be treated equally, so any operation you perform on one side must also be done on the other side to maintain balance. We start by aiming to get all the variables on one side and constants on the other. This often involves operations like addition, subtraction, multiplication, or division.In practice, this process helps to simplify the equation gradually until the solution is clearly seen. In our equation, adding 8 to both sides allowed the constant \(-8\) to be canceled from one side, effectively starting to solve the equation step by step.

Isolating Variables

Isolating variables is a crucial step in solving equations. The goal is to have the variable by itself on one side of the equation, making it easy to see what it equals.
In the equation \( \frac{1}{2}x - 8 = 1 \), the term with the variable \( \frac{1}{2}x \) is initially accompanied by a constant, which is \(-8\). To isolate \( \frac{1}{2}x \), we add 8 to both sides of the equation:

• This step eliminates the \(-8\), simplifying the equation to \( \frac{1}{2}x = 9 \).

Next, to fully isolate \(x\), we need to get rid of the fraction \( \frac{1}{2} \). This is done by multiplying both sides by 2. The concept here is to use the inverse operation to cancel the fraction:

• Multiplying \( \frac{1}{2} \) by 2 results in 1, effectively leaving \( x \) by itself.
• The operation on the other side—\( 2 \times 9 \)—gives us 18.

Now, \(x = 18\) with the variable isolated, offering a clear, final answer.

Verifying Solutions

Verifying solutions is a way to ensure that the solution we found is accurate. It's an important final step to confirm our answer.To verify, we substitute our solution back into the original equation. In this case, substitute \(x = 18\) into \( \frac{1}{2}x - 8 = 1 \). Performing the calculations gives us:

• Calculate \( \frac{1}{2} \times 18 \), which equals 9.
• Then subtract 8 from 9, resulting in 1.

Since both sides of the equation are equal—a balance of 1—it indicates that the solution \(x = 18\) is correct.
Verification is a simple yet effective check to solidify your understanding and confirm you've got the right answer.