Problem 5 - Elementary Algebra

Question

Solve the equations and inequalities for the following problems. $$ \frac{y-9}{4}+6=3 $$

Step-by-Step Solution

Answer

Answer: The value of y in the given equation is -3.

Step 1: Write down the given equation

We are given the equation: $$ \frac{y-9}{4}+6=3 $$

Step 2: Eliminate the fraction

In order to eliminate the fraction, we need to multiply each side of the equation by the denominator of the fraction, which is 4: $$ 4 \cdot \frac{y-9}{4} + 4 \cdot 6 = 4 \cdot 3 $$ This simplifies to: $$ y-9 + 24 = 12 $$

Step 3: Simplify both sides of the equation

Now, combine the constants on the left side of the equation: $$ y + 15 = 12 $$

Step 4: Solve for y

To solve for y, we need to isolate it on one side of the equation. We will subtract 15 from both sides of the equation: $$ y = 12 - 15 $$ This simplifies to: $$ y = -3 $$

Step 5: Write down the final solution

After following the steps above, we find that the solution to the equation is: $$ y = -3 $$

Key Concepts

Linear Equations

Linear equations are foundational in algebra and a key topic you'll encounter in mathematics. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. In essence, they express a relationship where one variable is dependent on the other in a direct way.
Examples of linear equations include:

• \(y = mx + b\)
• \(2x + 3 = 7\)

In the context of our original exercise, we worked with the equation \(\frac{y-9}{4} + 6 = 3\), which is linear in nature. Linear equations, like this one, can be solved with logical steps that simplify and isolate the variable.

Fraction Elimination

When dealing with linear equations, you might encounter fractions, as seen in the equation \(\frac{y-9}{4} + 6 = 3\). Dealing with fractions can seem tricky, but fraction elimination helps simplify such problems.
To eliminate a fraction:

• Identify the denominator of the fraction.
• Multiply every term in the equation by this denominator.

This method clears the fraction, making the equation easier to handle. In our example, multiplying both sides by 4 gave us \(y - 9 + 24 = 12\), effectively removing the fraction. This technique is crucial for keeping equations manageable and set the stage for solving the problem.

Variable Isolation

Isolating the variable is a fundamental step when solving equations. Variable isolation involves getting the variable of interest, like \(y\) in our exercise, on one side of the equation by itself.
Steps to isolate a variable include:

• Add or subtract terms to move them from one side to the other.
• Use opposite operations to cancel out numbers in front of the variable.

In our example, after simplifying the equation to \(y + 15 = 12\), subtracting 15 from both sides resulted in the isolated variable \(y = -3\). This step is crucial because it directly leads to finding the solution to the equation.

Mathematical Simplification

Simplification makes an equation easier to read and solve. In our example, after eliminating the fraction and combining like terms, we reached the simplified equation \(y + 15 = 12\).
The importance of simplification lies in:

• Reducing complex expressions into manageable parts.
• Combining constants or like terms whenever possible.

Simplifying expressions is not just about reaching a final number but about making calculations easier and reducing the chance for errors. The process prepares you for the final steps of solving, such as isolating variables, ensuring a clear path to finding a solution.