Problem 44 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$y+\frac{7}{11}=\frac{7}{11}$$

Step-by-Step Solution

Answer

The solution to the equation \(y + \frac{7}{11} = \frac{7}{11}\) is \(y = 0\).

Step 1: Identify the Equation

The first step is to identify the equation which is \(y + \frac{7}{11} = \frac{7}{11}\).

Step 2: Solve for y

Using the addition property of equality, subtract \(\frac{7}{11}\) from both sides of the equation: \(y + \frac{7}{11} - \frac{7}{11} = \frac{7}{11} - \frac{7}{11}\). This simplifies to \(y = 0\).

Step 3: Check the Solution

Verify that \(0\) is indeed a solution of the original equation: \[0 + \frac{7}{11} = \frac{7}{11}\]. It clearly holds true, showing that \(0\) is the solution to the equation.

Key Concepts

Solving Linear Equations

Linear equations are equations that involve variables, like \( y \), with no exponents or powers besides one. Solving linear equations is about finding the value of the variable that makes the equation true. Luckily, they are generally straightforward to work with using basic algebra concepts. The goal is to isolate the variable on one side of the equation.
\ In our problem \( y + \frac{7}{11} = \frac{7}{11} \), we want to solve for \( y \). To do this, we use the addition property of equality, a fundamental principle that allows us to add or subtract the same value from both sides of an equation without changing its truth.
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• We start by subtracting the same fraction \( \frac{7}{11} \) from each side:
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\ \[ y + \frac{7}{11} - \frac{7}{11} = \frac{7}{11} - \frac{7}{11} \]
\ This will leave us with \( y = 0 \). By carefully using this property, we can keep the balance of the equation while finding our solution.

Checking Solutions

Once we've found what we think is the solution, it's crucial to check it. This means substituting the solution back into the original equation to ensure it makes a true statement. This process can help us catch any errors we might have made during our calculations.
\ For the given equation \( y + \frac{7}{11} = \frac{7}{11} \), we substitute \( y = 0 \) back into the equation:
\ \[ 0 + \frac{7}{11} = \frac{7}{11} \]
\ This statement is clearly true, confirming that \( y = 0 \) is indeed the correct solution.
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• This step of checking helps reinforce our understanding and gain confidence in our solution.
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• It also serves as a safety net against calculation mistakes.
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Basic Algebra Concepts

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In basic algebra, understanding and applying certain concepts is key to solving equations effectively. Concepts like variables, operations, and properties form the foundation.
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• Variables: Symbols such as \( y \) that represent unknown values.
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• Operations: Includes addition, subtraction, multiplication, and division.
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• Properties of Equality: Such as the addition property, which allows us to maintain balance in an equation.
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\ These basic concepts work together to simplify problem-solving. Whether dealing with simple equations like \( y + \frac{7}{11} = \frac{7}{11} \) or more complex situations, understanding these principles will help guide you through solving and validating your solutions effectively. Always remember, practice in using these concepts enhances your problem-solving skills in algebra!