Problem 56 - Algebra 2

Question

Solve each equation. Check your solution. $$ \frac{3}{11} a-1=\frac{7}{11} a+9 $$

Step-by-Step Solution

Answer

The solution is \( a = -\frac{55}{2} \).

Step 1: Isolate Variable Terms

First, we need to get all the terms with the variable "a" on one side of the equation. Subtract \( \frac{7}{11} a \) from both sides: \[ \frac{3}{11} a - \frac{7}{11} a - 1 = 9 \] Simplify the left side: \[ -\frac{4}{11} a - 1 = 9 \]

Step 2: Simplify Constant Terms

Add 1 to both sides to isolate the term with "a": \[ -\frac{4}{11} a = 10 \]

Step 3: Solve for a

Multiply both sides by \( -\frac{11}{4} \) to solve for "a": \[ a = 10 \times -\frac{11}{4} \] Simplify the multiplication: \[ a = -\frac{110}{4} \] Reduce the fraction: \[ a = -\frac{55}{2} \]

Step 4: Check the Solution

Substitute \( a = -\frac{55}{2} \) back into the original equation to verify: Left side: \[ \frac{3}{11} \left(-\frac{55}{2}\right) - 1 = -\frac{165}{22} - 1 = -\frac{75}{11} - 1 \] Simplify: \[ -\frac{75}{11} - \frac{11}{11} = -\frac{86}{11} \] Right side: \[ \frac{7}{11} \left(-\frac{55}{2}\right) + 9 = -\frac{385}{22} + \frac{198}{22} = -\frac{187}{22} \approx -\frac{86}{11} \] Both sides are equal, confirming \( a = -\frac{55}{2} \) is correct.

Key Concepts

Isolation of Variables

One of the key techniques for solving linear equations is isolating the variable of interest, which in this case is "a". In the equation \( \frac{3}{11} a - 1 = \frac{7}{11} a + 9 \), the aim is to gather all terms involving the variable \(a\) on one side. This lets us focus on adjusting the equation to find the variable's value. Here’s how you do it:

• First, subtract \( \frac{7}{11} a \) from both sides of the equation to remove it from the right side. This gives \( \frac{3}{11} a - \frac{7}{11} a - 1 = 9 \).
• Simplify to \( -\frac{4}{11} a - 1 = 9 \).
• Add 1 to both sides to isolate \( -\frac{4}{11} a \), obtaining \( -\frac{4}{11} a = 10 \).

Now, the variable "a" is isolated and the problem is narrowed down to solving this simpler equation.

Checking Solutions in Equations

Once you've found a solution, it’s crucial to check if it satisfies the original equation. This helps verify that no errors were made during calculation. To check the solution \(a = -\frac{55}{2}\), substitute it back into the original equation:

• Substitute into left side: \( \frac{3}{11} \left(-\frac{55}{2}\right) - 1 = -\frac{165}{22} - 1 \).
• Convert \(-1\) to \( -\frac{11}{11} \) and combine to \( -\frac{75}{11} - \frac{11}{11} = -\frac{86}{11} \).
• Substitute into right side: \( \frac{7}{11} \left(-\frac{55}{2}\right) + 9 = -\frac{385}{22} + \frac{198}{22} = -\frac{187}{22} \).
• Both sides simplify to essentially match \(-\frac{86}{11}\), confirming the solution is correct.

This checking process ensures accuracy and understanding of the solution.

Fraction Arithmetic

A key skill in solving linear equations, especially with fractions, is fraction arithmetic. When you manipulate equations, you often need to perform operations involving fractions. In our example, several fraction operations were necessary:

• Subtracting like-fractions: \( \frac{3}{11} a - \frac{7}{11} a = -\frac{4}{11} a \).
• Multiplying by a fraction: Solving \( -\frac{4}{11} a = 10 \) requires multiplying both sides by \( -\frac{11}{4} \).
• Simplifying fractions: When calculations yield complex fractions, reduce them for simplicity. For instance, \( a = -\frac{110}{4} \) simplifies to \( a = -\frac{55}{2} \).

Understanding how to perform these operations helps eliminate mistakes and ensures the solution process is fluid.

Equation Simplification

Simplifying both sides of an equation is crucial for solving it efficiently. The main aim is to reduce the complexity of expressions, making it easier to isolate variable terms. Here’s a closer look:

• Simplification involves combining like terms. For example, gathering all "a" terms together by subtracting them from both sides.
• Altering complex expressions: Once terms are moved, equations like \( -\frac{4}{11} a - 1 = 9 \) need simplification by adding or subtracting numbers to isolate the variable term.
• Final simplification: Upon reaching \( -\frac{4}{11} a = 10 \), multiplying by \(-\frac{11}{4}\) is necessary to find "a" efficiently.

Equation simplification minimizes errors and enhances clarity throughout the problem-solving process.