Problem 11 - Algebra and Trigonometry

Question

\(9- 46\) 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: Understand the Equation

The given equation is \( \frac{1}{2}x - 8 = 1 \). Our objective is to isolate \( x \) on one side to solve for its value. This equation is linear as it can be simplified to the form of \( ax + b = c \).

Step 2: Add 8 to Both Sides

To start simplifying the equation, add 8 to both sides so we can eliminate the -8 from the left side. This yields: \( \frac{1}{2}x = 1 + 8 \), which simplifies to \( \frac{1}{2}x = 9 \).

Step 3: Multiply by 2 to Clear the Fraction

To eliminate the fraction, multiply both sides of the equation by 2. This results in: \( x = 9 \times 2 \).

Step 4: Simplify the Equation

Calculate \( 9 \times 2 \) to find \( x \). We get \( x = 18 \). The solution to the equation is \( x = 18 \).

Key Concepts

Isolating Variables

When solving linear equations, the goal is often to isolate the variable to find its value. In our exercise, the variable in question is \( x \). To isolate \( x \), we need it to be alone on one side of the equation.
This often involves using operations like addition, subtraction, multiplication, or division to "move" other terms.
Consider the equation \( \frac{1}{2}x - 8 = 1 \). Here, \(-8\) is hindering \( x \) from being by itself.
By adding 8 to both sides, we cancel it out on the left side, moving us closer to isolating \( x \).

• Perform opposite operations to eliminate terms. For example, if there's subtraction, add the same amount to both sides.
• Check your work to ensure each operation maintains the equation's balance.

This strategy is central to finding any unknown value effectively.

Equations with Fractions

Equations that contain fractions can often seem challenging, but they can be simplified just as easily as regular equations.
The key is to eliminate the fractions at the earliest point possible to simplify calculations and reduce errors.
In our exercise, the equation \( \frac{1}{2}x = 9 \) involves a fractional coefficient.
To get rid of this fraction, multiply both sides of the equation by 2, which is the denominator of the fraction.
This clears the fraction, transforming \( \frac{1}{2}x \) into \( x \).

• Identify the fraction in your equation.
• Multiply both sides by the denominator to clear the fraction.
• Solve the resulting equation as a normal linear equation.

Understanding this method greatly simplifies equations with fractions.

Simplifying Equations

Simplifying equations is a crucial step in solving linear equations efficiently.
It involves combining like terms and using basic arithmetic operations to make the equation as simple as possible for solving.
In the problem \( \frac{1}{2}x - 8 = 1 \), we simplified first by isolating \( x \) and then by removing the fraction.
This leaves us with a straightforward multiplication to solve: \( x = 9 \times 2 \).
Simplifying is not only about operations but also verifying that each step logically follows the previous one.

• Combine like terms where possible to reduce complexity.
• Ensure each transformation of the equation is accurate to avoid errors.
• Simplify systematically to check your work easily.

By consistently simplifying, solving equations becomes less daunting and more manageable.