Problem 44 - Fundamentals of Mathematics

Question

Calculator Exercises. $$y-2.161=5.063$$

Step-by-Step Solution

Answer

The solution is \( y = 7.224 \).

Step 1: Understanding the Equation

The given equation is a simple linear equation that needs to be solved for \( y \). The equation is \( y - 2.161 = 5.063 \).

Step 2: Isolate the Variable

Add 2.161 to both sides of the equation to isolate \( y \) on the left side: \( y - 2.161 + 2.161 = 5.063 + 2.161 \).

Step 3: Simplify the Equation

After adding 2.161 to both sides, the equation simplifies to: \( y = 5.063 + 2.161 \).

Step 4: Calculate the Result

Perform the addition on the right side: \( 5.063 + 2.161 = 7.224 \).

Step 5: Present the Solution

The value of \( y \) that satisfies the equation is \( y = 7.224 \).

Key Concepts

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. In the given exercise, we aim to find the value of \( y \) that satisfies the equation \( y - 2.161 = 5.063 \). By isolating \( y \), we can discover what value it must take for both sides of the equation to remain equal.
To solve, we perform operations to both sides, keeping the equation balanced, until the variable is isolated. The goal is to end up with \( y = \) something, where "something" is the result from our operations. Thus, solving equations relies on maintaining equality through careful manipulation.

Isolation of Variables

Isolation of variables is crucial when solving equations. It refers to getting the unknown variable, in this case \( y \), alone on one side of the equation. In our example, the equation \( y - 2.161 = 5.063 \) needs us to isolate \( y \) on the left side.
To isolate \( y \), we need to eliminate the -2.161. We achieve this by performing an arithmetic operation - adding 2.161 to both sides:

• Left Side: \( y - 2.161 + 2.161 = y \)
• Right Side: \( 5.063 + 2.161 \)

This operation effectively isolates \( y \), simplifying our equation to \( y = 7.224 \). It's akin to peeling away layers to reveal only what we want: the variable \( y \) by itself.

Arithmetic Operations

Arithmetic operations include actions like addition, subtraction, multiplication, and division. They are fundamental tools used in manipulating equations to solve them. In our exercise, addition is the key operation we use to isolate the variable \( y \).
After rewriting the original equation \( y - 2.161 = 5.063 \) to include our arithmetic operation, we perform the addition:

• Add 2.161 to both sides: \( y - 2.161 + 2.161 = 5.063 + 2.161 \)
• Simplify to find \( y = 7.224 \)

This straightforward arithmetic helps us transition from a complex-looking equation to a simple expression for \( y \). Always remember: performing the same operation on both sides maintains balance and is essential in solving for the variable efficiently.