Problem 73 - Algebra 2

Question

Solve each equation. $$ \frac{x}{2}+5=7 $$

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Identify and Isolate the Variable Term

The given equation is \(\frac{x}{2} + 5 = 7\). The goal is to isolate the variable term. First, subtract 5 from both sides of the equation: \(\frac{x}{2} + 5 - 5 = 7 - 5\). This simplifies to \(\frac{x}{2} = 2\).

Step 2: Eliminate the Fraction

Now, you need to eliminate the fraction so that \(x\) can stand alone. Multiply both sides of the equation by 2: \(2 \cdot \frac{x}{2} = 2 \cdot 2\). This simplifies to \(x = 4\).

Key Concepts

Variable Isolation

When solving linear equations, a crucial step is isolating the variable. This means separating the variable on one side of the equation to simplify the expression. Start by observing the equation carefully. Identify the variable and the operations surrounding it.

• In the given equation \(\frac{x}{2} + 5 = 7\), our goal is to have \(x\) by itself on one side.
• We do this by removing any additional numbers or operations. In this case, subtract 5 from both sides to maintain the balance.

This subtraction clears the constant from the equation, transforming it into \(\frac{x}{2} = 2\). Now the variable term is isolated, and we are ready to proceed to the next step. Remember, isolating the variable simplifies your work and aids in accurately solving the equation.

Fraction Elimination

Once the variable term is isolated in a fraction, the next step is to eliminate the fraction. This is essential to simplify the equation and make the variable stand alone.

• In the equation \(\frac{x}{2} = 2\), the variable is divided by 2.
• To eliminate the fraction, multiply both sides of the equation by 2.

This action scales up the equation, effectively canceling out the denominator on the left side. It results in a simpler equation of \(x = 4\). Multiplying by the reciprocal is a reliable method for fraction elimination. It balances the equation while freeing the variable from its fractional form.

One-Step Equations

After isolating the variable and eliminating fractions, solving turns into dealing with a one-step equation. This is usually the simplest form of an equation, with one operation left to perform.

• In our example, we started with a more complex equation but ended with the simple form \(x = 4\).
• There are no further operations needed, the variable solution is clear.

Solving one-step equations is often straightforward, making the earlier steps of variable isolation and fraction elimination worth the effort. This step confirms our work, letting us securely conclude that \(x = 4\) is indeed the solution. Understanding how to handle these simplest forms ensures accuracy in finding the solution.