Problem 11 - Algebra Form and Function

Question

Solve the equations. $$ \frac{2}{3}(3 n-12)=\frac{3}{4}(4 n-3) $$

Step-by-Step Solution

Answer

Answer: n = -5

Step 1: Simplify both sides

Simplify the left part of the equation by multiplying 2/3 with (3n-12), and the right part by multiplying 3/4 with (4n-3). $$ \frac{2}{3} (3n-12) = \frac{2}{1}(n-4) \\ \frac{3}{4} (4n-3) = \frac{3}{1}(n-1) $$

Step 2: Find the least common multiple of the denominators

The denominators are 3 and 4. The least common multiple of 3 and 4 is 12.

Step 3: Clear the fractions

Multiply both sides of the equation by 12, the least common multiple we found in Step 2. $$ 12 \times \frac{2}{1}(n-4) = 12 \times \frac{3}{1}(n-1) $$

Step 4: Simplify the equation

Perform the multiplication on both sides, and simplify the equation. $$ 12(2)(n-4) = 12(3)(n-1) \\ 24n - 96 = 36n - 36 $$

Step 5: Solve for n

Isolate n on one side of the equation by subtracting 24n from both sides, and then adding 96 to both sides. $$ 24n - 96 - 24n = 36n - 36 - 24n \\ -96 = 12n - 36 \\ -60 = 12n $$ Now, divide both sides by 12 to find the value of n. $$ \frac{-60}{12} = n \\ n = -5 $$ The solution is n = -5.

Key Concepts

Algebra and Its Role in Equation Solving

Algebra is a branch of mathematics that uses symbols to represent numbers in equations. It allows us to solve problems involving unknown values.
These symbols, often letters like \(n\) in our exercise, let us express general relationships. When tackling algebraic equations, such as fractions with variables, it's important to first understand what each part of the equation represents.
This gives us the groundwork to manipulate and solve for the unknown. Algebraic problem-solving helps build critical thinking and problem-solving skills, which are valuable in both mathematics and real-world applications. In the exercise provided, we are required to simplify and manipulate both sides of the equation to isolate and determine the value for \(n\). This is a typical method used in algebra to find solutions to equations.

Understanding and Solving Linear Equations

Linear equations, like the one in our exercise, are equations that involve variables raised to the power of one.
The main goal in solving them is to isolate the variable on one side to determine its value. In our given equation, both sides were simplified and set equal to one another. This is a key part of solving equations:

• Simplify each side of the equation individually.
• Use techniques like finding a common denominator or multiplying through by the least common multiple to clear fractions.
• Reorganize the equation as necessary to isolate the variable.

The process simplifies the equation to a point where it can be solved using basic arithmetic.

Dealing with Fractions in Equations

Fractions add a layer of complexity to equation solving, but they can be managed with a few strategies.
Often, fractions are part of problems because they represent a part of a whole. To solve equations involving fractions, it's often useful to eliminate them early in the process. Consider these steps for handling fractions:

• Identify the least common multiple (LCM) of the denominators to help eliminate the fractions by multiplication.
• Multiply every term in the equation by this LCM to remove the fractions.
• Be sure to simplify as you go, maintaining balance across the equation to not change its equality.

In our example, multiplying both sides of the equation by 12, the LCM of 3 and 4, removed the fractions. This conversion simplifies calculations and reduces errors.

Going Through Step-by-step Solutions

A step-by-step approach is crucial for understanding and solving complex problems.
This methodical breakdown is designed to help identify errors early and ensure each part of the process is clear. Here's how the step-by-step approach works:

• Simplification: Start by simplifying each side of the equation as much as possible.
• Balancing Equations: Always keep the equation balanced by performing any operation equally on both sides.
• Isolation of the Variable: Aim to have the variable on one side with all other terms on the opposite side.
• Final Calculation: Solve to find the value of the variable using basic operations like addition, subtraction, multiplication, or division.

Following these steps thoroughly helps in arriving at the correct solution, as illustrated in the example where the solution \(n = -5\) was reached by carefully following each step.