Problem 53 - Elementary Algebra

Question

Solve \(4 y+2 x+8=0\) for \(y\).

Step-by-Step Solution

Answer

Answer: \(y = \dfrac{-(x + 4)}{2}\)

Step 1: Rewrite the equation

We can start by rewriting the equation as follows: \(4y + 2x + 8 = 0\).

Step 2: Isolate the term with the variable y

To isolate the term with the variable \(y\), we will subtract both \(2x\) and \(8\) from both sides of the equation: \(4y = -2x - 8\).

Step 3: Solve for y

Finally, to solve for \(y\), we will divide both sides of the equation by 4: \(y = \dfrac{-2x - 8}{4}\).

Step 4: Simplify the expression

We can simplify the expression by factoring out a common factor of 2 from the numerator: \(y = \dfrac{-2(x + 4)}{4}\). Then, we can reduce the fraction by dividing both numerator and denominator by 2: \(y = \dfrac{-1(x + 4)}{2}\).

Step 5: Write the final answer

After simplifying the expression, we have the final answer: \(y = \dfrac{-(x + 4)}{2}\).

Key Concepts

Solving Equations

At the heart of solving equations like the one given in the exercise, the primary goal is to find the value of one variable in terms of the others. In the provided equation, which is \(4y + 2x + 8 = 0\), we are focusing on determining \(y\). To solve such equations, the first step is rearranging terms:

• Identify the variable you need to solve for, in this case, \(y\).
• Perform operations on both sides of the equation to isolate this variable. For instance, subtract components not involving \(y\) from both sides.

By doing these, you can successfully isolate the desired variable. A structured approach, methodical manipulation, and performing the same operation on both sides are key in maintaining the equation's equality throughout the process.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging and transforming equations into simpler or more useful forms. It involves:

• Adding, subtracting, multiplying, or dividing both sides of an equation by the same number or expression.
• Using inverse operations to handle components of an equation, such as reversing addition with subtraction, or vice versa.

In the original exercise, after isolating the term \(4y\), the next step was to eliminate the coefficient next to \(y\) by dividing both sides by 4. This is a common algebraic manipulation tactic to solve for a variable that has a coefficient. The art comes in knowing which operation to perform to simplify solving the equation.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that involves making an expression more compact or easier to understand. This can be done by:

• Factoring common terms from expressions.
• Reducing fractions to their simplest form.

In this practice, the expression \(\dfrac{-2x - 8}{4}\) was simplified by factoring out a common factor of 2 from the numerator, resulting in \(\dfrac{-2(x + 4)}{4}\). Further simplification was achieved by dividing both numerator and denominator by 2, leading to the expression \(\dfrac{-(x + 4)}{2}\). Such simplifications are crucial as they provide clearer and more manageable forms, making it easier to interpret and understand the solution.