Problem 3 - Elementary Algebra

Question

Solve \(\frac{y}{8}=-2\) for \(y\)

Step-by-Step Solution

Answer

Answer: y = -16

Step 1: Write down the given equation

We are given the equation: \(\frac{y}{8}=-2\)

Step 2: Multiply both sides by 8 to isolate \(y\)

To eliminate the denominator, multiply both sides by 8: \(8 \times \frac{y}{8} = -2 \times 8\) This simplifies to: \(y = -16\)

Step 3: Write the final answer

The solution to the equation is \(y = -16\).

Key Concepts

Isolate the Variable

When it comes to solving linear equations, a fundamental goal is to isolate the variable - in other words, to get the variable by itself on one side of the equation. This process allows us to find the value of the variable that makes the equation true.

For instance, if you have an equation like \( \frac{y}{8} = -2 \), your variable \( y \) is not alone; it's currently part of a fraction. To isolate \( y \) and solve for it, we look to eliminate any coefficients or terms that are attached to \( y \). In the provided example, \( y \) is divided by 8, so to isolate it, we do the opposite operation: multiplying both sides of the equation by 8. This cancels out the division by 8 on the left side, leaving \( y \) isolated.

Isolating the variable often involves basic arithmetic operations: adding, subtracting, multiplying, or dividing both sides of the equation by the same number. Remember, whatever you do to one side of the equation, you must do to the other to keep the equation balanced.

Algebraic Fractions

Equations can sometimes include algebraic fractions, which are fractions that contain one or more variables. In the equation \( \frac{y}{8} = -2 \), the algebraic fraction is \( \frac{y}{8} \). Algebraic fractions can seem daunting at first, but they follow the same principles as numerical fractions.

When solving equations with algebraic fractions, one of the main objectives is to eliminate the fractions so that you can work with whole numbers. This often simplifies the equation and makes it much easier to solve. To remove an algebraic fraction, you can multiply each term in the equation by the least common denominator (LCD) or, in simpler cases like our example, by the denominator of the fraction you are trying to eliminate. The key is to convert the equation to a simpler form without fractions before proceeding with other steps to solve for the variable.

Equation Solving Steps

The process of equation solving involves a few essential steps, which can be applied to a wide variety of problems. The first step is to understand the given equation and identify the variable you're solving for. Next, simplify the equation by clearing any fractions or combining like terms, if necessary.

Once simplified, perform operations that will isolate the variable. This may involve moving terms from one side of the equation to the other by adding, subtracting, multiplying, or dividing both sides accordingly. It's crucial to maintain balance by performing the same operation on both sides of the equation.

Finally, once the variable is isolated, you can find its value. Always check your solution by substituting it back into the original equation to verify that it satisfies the equation. This step-by-step methodical approach helps ensure accuracy and builds a solid foundation for tackling more complex algebraic problems.