Problem 40 - Elementary Algebra

Question

Solve the equations. $$ \frac{y-21}{8}=-3 $$

Step-by-Step Solution

Answer

Answer: y = -3

Step 1: Eliminate the fraction

To eliminate the fraction, we multiply both sides of the equation by the denominator (8): $$ 8\left(\frac{y-21}{8}\right) = -3 \cdot 8 $$

Step 2: Simplify the equation

Now, we can simplify the equation by canceling out the 8's on the left side and performing the multiplication on the right side: $$ y - 21 = -24 $$

Step 3: Solve for y

To solve for y, we need to isolate the variable by adding 21 to both sides of the equation: $$ y - 21 + 21 = -24 + 21 $$ This simplifies to: $$ y = -3 $$ So, the solution to the given equation is y = -3.

Key Concepts

Linear Equations

Linear equations are equations where the highest power of the variable is one. They are fundamental in algebra and are generally in the form of

• The famous form: \(ax + b = 0\) and
• The common type: \(y = mx + c\).

In the problem given, our goal is to find the value of \(y\) such that both sides of the equation are balanced. We often encounter linear equations in everyday scenarios, like determining costs, distances, or even time. Understanding how to manipulate these equations is crucial for grasping more advanced algebraic concepts.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. The process includes the following:

• Identify the variable you need to solve.
• Use operations to isolate the variable on one side.

In the provided solution, solving the equation started with eliminating the fraction, then simplifying both sides until \(y\) stood alone on one side.
Key operations include addition, subtraction, multiplication, and division, as these help to rearrange the terms. Always remember to perform the same operation on both sides of the equation to maintain balance.

Fraction Elimination

A common technique in solving equations with fractions is fraction elimination. This is crucial because working with whole numbers is generally simpler than working with fractions.
Here's how it helps:

• Multiply every term by the denominator to remove the fraction.
• This reduces the complexity of the equation.

In the given equation, multiplying both sides by 8 removed the fraction, allowing us to focus on solving a much simpler linear equation: \(y - 21 = -24\). Removing fractions early on makes it easier to solve for the variable since it leads to straightforward arithmetic operations.