Problem 19 - Algebra 1

Question

Solve the equation. $$\frac{x}{2}+13=20$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 14\).

Step 1: Isolate the term with x

In this equation, the term with the variable \(x\) is \(\frac{x}{2}\). Begin with isolating this term by subtracting 13 from both sides of the equation which will give: \(\frac{x}{2}+13-13=20-13\) that simplifies to \(\frac{x}{2}=7\).

Step 2: Solve for x

Now that we have isolated the term with \(x\), we can solve for \(x\) by multiplying both sides of the equation by 2. This gives \(2 * \frac{x}{2}= 7 * 2\) which simplifies to \(x = 14\).

Key Concepts

Isolating Variables

One of the foundational skills in solving linear equations is isolating the variable. Here, the variable to be isolated is \( x \). To do this, focus on removing any other terms or constants from its side of the equation.

In the equation \( \frac{x}{2} + 13 = 20 \), the term \( \frac{x}{2} \) needs to be isolated. Start by eliminating the constant 13 from the left side:

• Subtract 13 from both sides of the equation.

By doing this, the equation simplifies to \( \frac{x}{2} = 7 \).

This clear separation of the variable term is a crucial step. It allows you to focus solely on the term containing \( x \) for further simplification.

Balancing Equations

Balancing an equation is essential to maintain equality. When you manipulate one side of the equation, you must do the same to the other side.

In our equation, subtracting 13 is done to both sides. This keeps the equation balanced and maintains equality. It's akin to balancing a scale - if you remove or add weight to one side, the same must be done to the other.

• Always perform the same operation on both sides.
• This maintains the equation's equivalence.

Keeping equations balanced prevents errors and ensures that the solution you find is accurate.

Elementary Algebra

Elementary algebra serves as the foundation for solving equations like \( \frac{x}{2} + 13 = 20 \). It encompasses various techniques, including isolating variables and performing operations.

Let's solve for \( x \) in the equation. Once the term \( \frac{x}{2} \) is isolated, proceed to eliminate the fraction by multiplying both sides by 2. This step transforms the equation:

• Multiply both sides: \( 2 \times \frac{x}{2} = 7 \times 2 \)

This cancels out the fraction to simply become \( x = 14 \).

This straightforward approach highlights key principles of elementary algebra, such as dealing with fractions and performing arithmetic operations.