Problem 8 - Prealgebra

Question

Solve each equation. $$-7 y=-0.63$$

Step-by-Step Solution

Answer

The solution is \(y = 0.09\).

Step 1: Identify the Equation

The given equation is \(-7y = -0.63\). This is a linear equation where \(y\) is the variable we need to solve for.

Step 2: Isolate the Variable

To solve for \(y\), we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by \(-7\):\[y = \frac{-0.63}{-7}\].

Step 3: Simplify the Right Side

Calculate the division on the right-hand side to find the value of \(y\):\[y = \frac{-0.63}{-7} = 0.09\].

Key Concepts

Variable Isolation

Variable isolation is an essential step when solving linear equations. Linear equations typically are in the form of \(Ax + B = C\), where you have one variable to solve for. In our given exercise, the equation \(-7y = -0.63\) needs us to find \(y\). The goal is to "isolate" the variable, meaning we want \(y\) to stay alone on one side of the equation.To isolate \(y\), we perform operations that will not change the equality of the equation. This involves using inverse operations to "cancel out" or "clear" terms around \(y\). In our example, since \(-7\) is multiplied by \(y\), the inverse operation will be division. When we divide both sides of the equation by \(-7\), we effectively remove the \(-7\) from the left side, leaving us with:\[y = \frac{-0.63}{-7}\].These operations keep the equation balanced, leading us directly to finding the value of \(y\). Remember, when isolating the variable, it's imperative to perform the same operation on both sides to maintain equality.

Solving Equations Step-by-Step

Solving equations step-by-step helps us to systematically find the solution without confusion. By breaking down each action into clear, logical steps, we avoid mistakes and ensure we understand why each step is necessary. Let’s break down the steps for solving linear equations:

• **Identify the equation's structure:** Look for the term with the variable and constants.
• **Perform inverse operations:** Use inverse mathematical operations to move terms from one side of the equation to the other. Ensure to do identical operations on both sides to keep the equation balanced.
• **Simplify each step:** After each operation, simplify the equation as needed, which often involves combining like terms or reducing fractions.

In our example, we first identified the linear equation \(-7y = -0.63\). Then, we isolated \(y\) by performing the division. The final step was simplifying that division to find \(y = 0.09\). By following these logical steps, we reach the solution efficiently and accurately.

Simplifying Fractions

Simplifying fractions is about finding an equivalent fraction in its simplest form. A fraction is simplified when the numerator and denominator have no common factors other than 1. Doing this makes calculations and interpretations much easier.In the exercise, we had to simplify \(\frac{-0.63}{-7}\). Simplifying a division like this usually involves performing basic division to see if the numerator can be evenly divided by the denominator. In this scenario:\[\frac{-0.63}{-7} = 0.09\].By calculating the division, we ended up with a simple decimal value. In general, simplification can also involve reducing larger fractions through methods like finding the greatest common divisor (GCD). However, with our specific problem, the direct division simplified the fraction immediately. Understanding how to simplify fractions correctly ensures that your equations’ solutions are as precise and concise as possible.