Problem 6 - Intermediate Algebra

Question

Given the equation \(3 x+y=-4\) a. solve for \(x\) b. solve for \(y\) c. Which variable was easier to solve for? Explain why.

Step-by-Step Solution

Answer

Solving for \(y\) is easier because it requires fewer steps.

Step 1: Solving for x

To solve for \(x\) in the equation \(3x + y = -4\), we need to isolate \(x\) on one side of the equation. Start by subtracting \(y\) from both sides of the equation: \[3x = -4 - y\] Next, divide both sides by 3 to solve for \(x\): \[x = \frac{-4 - y}{3}\].

Step 2: Solving for y

To solve for \(y\) in the equation \(3x + y = -4\), isolate \(y\) on one side. This can be done by subtracting \(3x\) from both sides of the equation: \[y = -4 - 3x\].

Step 3: Determine Which Variable is Easier to Solve

Solving for \(y\) was easier because it involved only one step: subtracting \(3x\). Solving for \(x\) required two steps: first isolating \(3x\) and then dividing by 3. This additional step makes solving for \(x\) slightly more complex.

Key Concepts

The Skill of Isolating Variables

Isolating variables is key when solving linear equations. This process means rearranging an equation to get a specific variable by itself on one side. The goal is to make this variable the subject of the formula. For example, when we look at the equation \(3x + y = -4\), and we want to solve for \(x\), we start by moving all other terms to the opposite side.
This involves:

• Subtracting \(y\) from both sides to get \(3x = -4 - y\).
• Then, divide every term by 3 to completely isolate \(x\) and find \(x = \frac{-4 - y}{3}\).

With these steps, you can clearly see how isolating a variable transforms the equation to express \(x\) solely in terms of other known parts. This method forms the foundation of solving algebraic equations efficiently.

Understanding Equation Manipulation

Equation manipulation is the art of keeping equations balanced while making them easier to solve. This involves performing the same operation on both sides of an equation to maintain equivalence. To illustrate, in the equation \(3x + y = -4\), let's focus on solving for \(y\).
To isolate \(y\), we subtract \(3x\) from both sides, resulting in \(y = -4 - 3x\).
By manipulating the equation in this way, we're redistributing its terms to separate \(y\) nicely on one side. The operations can include adding, subtracting, multiplying, or dividing by the same number across the equation. These methods ensure that whatever changes we make do not alter the original relationship.

Deciphering Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations, like \(3x + y\), seen in our equation. Understanding these expressions is crucial because they describe a relationship that you can manipulate to isolate variables or solve for unknowns.
Let's break down the expression \(3x + y\):

• The term \(3x\) shows us \(x\) is being multiplied by 3.
• The term \(y\) is a standalone variable, ready to be adjusted in relation to other parts of the equation.
• The expression as a whole represents a set relationship, to which operations like addition or subtraction can alter but not destroy.

When we adeptly manage algebraic expressions, we can simplify, isolate variables, and ultimately uncover solutions to complex problems with ease. Always remember, each part of the expression plays a role in finding a path to the answer.