Problem 36 - Elementary Algebra

Question

Solve the equations. $$ \frac{3 x}{4}+2=14 $$

Step-by-Step Solution

Answer

Answer: x = 16

Step 1: Subtract 2 from both sides

We start by subtracting 2 from both sides of the equation in order to remove the constant term: $$ \frac{3 x}{4} + 2 - 2 = 14 - 2 \\ \Rightarrow \frac{3 x}{4} = 12 $$

Step 2: Multiply both sides by the reciprocal of 3/4

To isolate x, we multiply both sides of the equation by the reciprocal of the fraction 3/4, which is 4/3: $$ \frac{4}{3} \cdot \frac{3 x}{4} = 12 \cdot \frac{4}{3} \\ \Rightarrow x = 16 $$ The final solution for the equation is x = 16.

Key Concepts

Solving Linear Equations

Linear equations are the simplest form of algebraic equations, characterized by variables raised to the first power. They can usually be written in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. Solving a linear equation means finding the value of the variable that makes the equation true. To do this, we typically isolate the variable on one side of the equation. Here are steps commonly followed when solving:

• Identify the variable: Determine which letter represents the unknown quantity you need to solve for.
• Isolate the variable: Use mathematical operations (addition, subtraction, multiplication, division) to get the variable alone on one side of the equation.
• Check your solution: Substitute the value back into the original equation to ensure it holds true.

Linear equations can include various terms and layers, but by following these steps, you can simplify and solve them easily.

Fractions in Equations

Fractions in equations might seem challenging initially, but they can be managed with just a bit of practice. When you encounter a fraction in an equation, the goal is often to remove the fraction in order to simplify your work. Here's how you can approach equations with fractions:

• Understand how to simplify fractions: Ensure the fraction is simplified by dividing both numerator and denominator by their greatest common divisor (GCD).
• Clear the fractions: Multiply every term in the equation by the least common denominator (LCD) to eliminate the fractions.
• Use the reciprocal: In cases where the fraction is connected with a variable (like \( \frac{3x}{4} \)), applying the reciprocal can help isolate the variable.

In our example, multiplying by the reciprocal of \(\frac{3}{4}\) was a crucial step to clearing the fraction, enabling us to isolate \(x\) effectively.

Step-by-Step Problem Solving

Step-by-step problem solving is an effective strategy for tackling algebra equations, particularly for beginners. By breaking down the problem into smaller, manageable parts, you can focus on one thing at a time. This reduces complexity and errors. Here’s how to apply this approach:

• Analyze the problem: Before jumping into calculations, take a moment to understand what is being asked. Identify what you need to find and what information is provided.
• Plan your steps: Determine the sequence of operations that will lead you to the solution. This may include adding, subtracting, multiplying, or dividing terms.
• Execute and verify: Carefully perform each step, checking your work as you go. Once completed, verify your solution by substituting it back into the original equation.

This method not only helps you with the current problem but also builds skills for more complex equations in the future.