Problem 31 - Algebra 1

Question

Solve the equation. $$\frac{9}{2}(x+3)=27$$

Step-by-Step Solution

Answer

The solution to the equation is x = 3.

Step 1: Isolate the variable

Let's start with the original equation \(\frac{9}{2}(x+3)=27\). To isolate 'x', first we need to get rid of the fraction. Multiply both sides by \( \frac{2}{9} \). The equation becomes: \(x+3 = \frac{27}{\frac{9}{2}} = 6\).

Step 2: Solve for x

Now that we have \(x+3 = 6\), we can solve for 'x' by subtracting 3 from both sides of the equation. This gives us: \(x = 6 - 3 = 3\).

Key Concepts

Algebra Basics

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations. It's like a puzzle where we find the value of unknown variables using certain rules. In algebra, variables such as \( x \) stand in for numbers, and we perform the same mathematical operations as we do with numbers: addition, subtraction, multiplication, and division.

• Symbols and Equations: Equations are like balanced scales, where both sides are equal. The goal is to find the value of the unknown variable that will make the equation true.
• Operations: We perform operations in a specific order, following rules like "PEMDAS", which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Understanding these basics is crucial for solving algebraic equations efficiently and accurately.

Isolating the Variable

Isolating the variable is a key strategy in solving equations and involves manipulating the equation to get the unknown variable by itself on one side of the equation. Think of it as unwrapping a present where layers of operations are like wrapping papers that you must remove to reach the gift (the variable).

• Removing Coefficients: Begin by simplifying the equation if fractions are present. Multiply by the reciprocal when needed, as seen here by multiplying both sides by \( \frac{2}{9} \) to get rid of the fraction.
• Handling Constants: Once all terms connected to the variable are simplified, apply inverse operations to eliminate constants. For example, subtracting 3 from both sides in the step \( x + 3 = 6 \).

By systematically applying these techniques, you "isolate" the variable to find its value.

Step-by-Step Solution

Solving an equation step-by-step ensures accuracy and provides a clear understanding of the process. Each step involves using algebraic manipulations to simplify the equation and eventually solve for the unknown variable.
Let's break down our original problem, \( \frac{9}{2}(x+3)=27 \):

• Step 1: Clear the Fraction: Multiply both sides by the reciprocal of the fraction, \( \frac{2}{9} \), so the left side simplifies to \( x+3 \), and the right side becomes 6.
• Step 2: Solve for the Variable: Eliminate any remaining constants by subtracting 3, which gives \( x = 3 \).

These systematic steps clarify the method used, making complex problems seem manageable and ensuring that solutions are accurate.