Problem 16 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ \frac{13}{11} y-\frac{2}{11} y=-3 $$

Step-by-Step Solution

Answer

The solution is \( y = -3 \).

Step 1: Simplify Each Side

First, we will simplify the left side of the equation by combining the like terms. The like terms are \( \frac{13}{11}y \) and \( -\frac{2}{11}y \). Subtract \( \frac{2}{11}y \) from \( \frac{13}{11}y \) to get \[ \frac{11}{11}y = 1y = y \]. So the equation simplifies to \( y = -3 \).

Step 2: Solution Verification

Now, substitute \( y = -3 \) back into the original equation to verify the solution. Compute the left side: \( \frac{13}{11}(-3) - \frac{2}{11}(-3) \). Calculate each separately: \( \frac{13}{11}(-3) = -\frac{39}{11} \) and \( \frac{2}{11}(-3) = -\frac{6}{11} \). Then, the subtraction becomes: \( -\frac{39}{11} - (-\frac{6}{11}) = -\frac{39}{11} + \frac{6}{11} = -\frac{33}{11} = -3 \). The left side equals the right side, confirming the solution is correct.

Key Concepts

Simplifying Expressions

When working with linear equations, one of the first steps we often take is to simplify expressions on both sides of the equation. Simplifying involves combining like terms and making the equation easier to solve. In the example provided, we started with the equation \( \frac{13}{11}y - \frac{2}{11}y = -3 \).

• We identify the like terms: \( \frac{13}{11}y \) and \( -\frac{2}{11}y \). "Like terms" are terms that have the same variable raised to the same power. In this case, both terms have the variable \( y \) raised to the power of 1.
• To simplify, we combine these like terms by performing the arithmetic operation (subtraction in this case). We subtract \( \frac{2}{11}y \) from \( \frac{13}{11}y \), resulting in \( \frac{11}{11}y \), which simplifies further to \( 1y \) or just \( y \).

Simplifying expressions not only makes the equation look neater, but it also helps in seeing the path to the solution more clearly. The main goal is reducing the equation to a simpler form that is easier to work with.

Checking Solutions

Once we think we've found the solution to an equation, the next crucial step is checking our work. Verification ensures that our solution satisfies the original equation. To check a solution, we substitute the value back into the original equation and ensure that both sides of the equation are equal.
Let's look at the provided solution: we simplified to \( y = -3 \). To verify, substitute \( -3 \) for \( y \) back into the left side of the original equation:

• Calculate \( \frac{13}{11}(-3) \), which gives us \( -\frac{39}{11} \).
• Calculate \( \frac{2}{11}(-3) \), which results in \( -\frac{6}{11} \).
• Perform the operation: \( -\frac{39}{11} - (-\frac{6}{11}) = -\frac{39}{11} + \frac{6}{11} = -\frac{33}{11} \).
• Simplify \( -\frac{33}{11} \) to \( -3 \), which matches the right side of the equation (\(-3\)).

By confirming the sides are equal, we have shown our solution is correct. Checking is an important part of the problem-solving process, ensuring accuracy in our computations.

Like Terms

Understanding the concept of "like terms" is key when simplifying algebraic expressions. Like terms are terms that contain the same variable raised to the same power. This includes both the variable's presence and its exponent. In the context of our problem, terms are considered "like" if they are associable in operations like addition or subtraction.
Here's why identifying like terms is crucial:

• It allows for proper simplification. In the equation \( \frac{13}{11}y - \frac{2}{11}y \), both terms involve the variable \( y \), enabling us to combine them through subtraction.
• Simplification reduces the complexity of equations. Combining \( \frac{13}{11}y \) and \( -\frac{2}{11}y \) helps us reduce multiple terms to a single term, \( y \), thereby streamlining our calculation.
• Correct identification of like terms avoids errors in solving. Failing to group like terms may lead to incorrect simplification and a wrong solution.

Recognizing and combining like terms is a fundamental skill in algebra that aids in transforming and solving equations efficiently. This step is often one of the first taken when approaching algebraic problems, leading to quicker and accurate results.