Problem 1 - Algebra A Combined Function

Question

Solve each equation and check each solution. See Examples 1 through 3. $$ \frac{x}{5}+3=9 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 30 \).

Step 1: Isolate the Variable Term

We start by moving the constant term to the other side of the equation. Given the equation \( \frac{x}{5} + 3 = 9 \), subtract 3 from both sides to isolate the fraction containing \( x \) on one side: \( \frac{x}{5} + 3 - 3 = 9 - 3 \), which simplifies to \( \frac{x}{5} = 6 \).

Step 2: Solve for the Variable

Next, we need to eliminate the fraction to solve for \( x \). To do this, multiply both sides of the equation by 5: \( 5 \times \frac{x}{5} = 6 \times 5 \). This simplifies to \( x = 30 \).

Step 3: Check the Solution

To ensure the correctness of our solution, substitute \( x = 30 \) back into the original equation: \( \frac{30}{5} + 3 = 9 \). Simplify the left side: \( 6 + 3 = 9 \). Since both sides of the equation are equal, the solution \( x = 30 \) is verified.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill for solving linear equations. It involves rearranging and simplifying equations using various mathematical operations. This skill is crucial to unlocking the power of algebra and finding solutions efficiently. Here’s a breakdown of steps typically involved:

• Identifying like terms: Group similar terms to simplify your equation.
• Using inverse operations: Apply the opposite operation to both sides of the equation to maintain balance. For instance, if you have addition on one side, use subtraction on both sides.
• Combining like terms: Simplify the equation further by combining similar terms.

In our original exercise, after identifying the terms, we use subtraction to eliminate the constant, thereby simplifying the equation to \( \frac{x}{5} = 6 \). This simplification sets the foundation for solving the equation easily and accurately.

Isolation of Variables

Isolation of variables is an essential step for solving equations. The goal is to get the variable you're solving for by itself on one side of the equation. This simplifies the process of finding the solution. Here's how you might isolate a variable:

• Remove constants: Move numbers without variables to the opposite side by using inverse operations.
• Eliminate coefficients: Divide or multiply to remove any coefficients attached to the variable.
• Clear fractions: Multiply the entire equation by the denominator to get rid of fractions.

In the exercise, after simplifying, you isolate \( x \) by multiplying both sides by 5. This removes the fraction, resulting in \( x = 30 \). Isolating the variable ensures we can clearly see what the variable equals.

Verification of Solutions

Verification of solutions is a crucial step to ensure the accuracy of your work. After finding the potential answer, substitute it back into the original equation to see if it creates a true statement. Here’s a guide to verify solutions:

• Substitute back: Replace the variable with your solution.
• Simplify: Perform operations to see if the equation holds true.
• Compare: The left side should equal the right side of the equation.

In our original problem, the solution \( x = 30 \) was substituted back into the starting equation, giving us \( \frac{30}{5} + 3 = 9 \). Since simplifying gives \( 9 = 9 \), the solution is verified as correct. This step confirms that the solution meets all the conditions of the original equation, providing confidence in your answer.