Problem 38 - Elementary Algebra

Question

Solve. $$ -58 y=-52 $$

Step-by-Step Solution

Answer

\(y = \frac{26}{29}\)

Step 1: Identify the Equation

The given equation is \(-58y = -52\). Our goal is to solve for the variable \(y\).

Step 2: Isolate the Variable

To isolate \(y\), divide both sides of the equation by \(-58\). This gives you: \[ y = \frac{-52}{-58} \].

Step 3: Simplify the Expression

Simplify \(\frac{-52}{-58}\) by cancelling out the negative signs and reducing the fraction. The negative signs cancel each other, giving \( \frac{52}{58} \). Further simplify the fraction by dividing the numerator and the denominator by 2, resulting in \( \frac{26}{29} \).

Step 4: Write the Final Answer

The simplified expression represents the value of \(y\). Thus, the value of \(y\) is \( \frac{26}{29} \).

Key Concepts

Solving for Variables

When faced with a linear equation, our main task is to find the value of the unknown variable. In this equation, we want to solve for the variable \( y \). This usually involves isolating the variable on one side of the equation. Here’s how you approach it:

• Examine the equation carefully. Identify the side of the equation where the variable resides.
• Dividing or multiplying both sides of the equation by the coefficient of the variable helps isolate the variable. The coefficient is the number directly in front of the variable.

In our exercise, the equation given is \( -58y = -52 \). To isolate \( y \), divide both sides by \(-58\). This results in \( y = \frac{-52}{-58} \). Isolating variables is a fundamental step in solving linear equations and is crucial for finding solutions.

Fraction Simplification

Simplifying fractions is an important skill in mathematics as it helps make expressions cleaner and more understandable. When simplifying fractions, the goal is to make the numerator (top of the fraction) and the denominator (bottom of the fraction) as small as possible while keeping the value of the fraction the same.In the equation \( y = \frac{-52}{-58} \):

• Firstly, notice the negative signs in the fraction. Two negatives make a positive, so \( \frac{-52}{-58} \) simplifies to \( \frac{52}{58} \).
• Next, look for a common factor of both numbers. Here, 2 is a common factor.
• Divide both the numerator and the denominator by 2 to further simplify the fraction, leading to \( \frac{26}{29} \).

This makes the fraction simpler and provides the final value of \( y \). Simplicity in math expressions not only makes solving problems easier but also reduces the potential for computational errors.

Algebraic Manipulation

Algebraic manipulation is a process used to transform equations into more workable or simplified forms. This is key when handling linear equations. It involves applying various mathematical operations—like addition, subtraction, multiplication, division, and factoring—to both sides of an equation.For the given linear equation \( -58y = -52 \):

• Identify the operation you need to perform to isolate the variable, in this case, dividing both sides by \(-58\).
• Ensure each operation maintains the balance of the equation; whatever you do on one side must be done on the other.
• After isolating the variable, further mathematical techniques, like reducing fractions, are applied to achieve the most simplified form of the variable's value.

Mastering algebraic manipulation allows you to break down complex problems into manageable parts and is essential for tackling higher-level mathematics problems.