Problem 11 - College Algebra with Corequisite Support

Question

For each of the following exercises, solve the equation for y in terms of \(x\). $$ 4 x+2 y=8 $$

Step-by-Step Solution

Answer

\( y = 4 - 2x \)

Step 1: Identify the Equation

We start with the given equation: \(4x + 2y = 8\). Our goal is to isolate \(y\) on one side of the equation to express \(y\) in terms of \(x\).

Step 2: Move the 'x' Term

Subtract \(4x\) from both sides of the equation to move the \(x\) term to the right. This will give us: \(2y = 8 - 4x\).

Step 3: Isolate 'y'

Divide each term in the equation \(2y = 8 - 4x\) by 2 to isolate \(y\). Performing the division, we get: \(y = \frac{8}{2} - \frac{4x}{2}\).

Step 4: Simplify

Simplify the equation \(y = \frac{8}{2} - \frac{4x}{2}\) by carrying out the division in each term. This results in: \(y = 4 - 2x\).

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics, often appearing in various forms and applications. These equations primarily take the form of a straight line when graphed in a two-dimensional coordinate system. The general structure of a linear equation in two variables is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. In our original exercise, the equation \(4x + 2y = 8\) follows this form. Here are some key points to remember about linear equations:

• They represent a straight line on a graph.
• Involve variables raised only to the first power.
• Have constant coefficients.

Understanding the basic format and characteristics of linear equations is crucial for solving problems involving these equations. Recognizing the structure helps in performing operations to isolate variables and simplify equations effectively.

Variable Isolation

Variable isolation is an essential step in solving equations, especially when you need to express one variable in terms of another. The primary goal is to manipulate the equation to get a single variable on one side. This often involves reversing operations. Let's delve into some strategies for isolating variables:

• Identify Operations: Determine the operations affecting the variable to be isolated. In our exercise, the initial goal was to isolate \(y\) in \(4x + 2y = 8\).
• Inverse Operations: Use inverse operations to move terms across the equation. For instance, subtraction is the inverse of addition, used to move \(4x\) in the example.
• Division or Multiplication: Once the variable term is isolated with its coefficient, use division or multiplication to solve for the variable. For instance, dividing both sides by 2 isolated \(y\).

By carefully choosing and applying these steps, isolating the desired variable becomes efficient, paving the way for further simplification.

Equation Simplification

Equation simplification involves making an equation easier to understand and solve by reducing it to its simplest form. This step is crucial after isolating the variable. Here's how it generally works:

• Combine Like Terms: If there are terms that can be combined, do so to simplify the expression.
• Perform Arithmetic Operations: Execute any necessary arithmetic, such as addition, subtraction, multiplication, or division, to simplify fractions. In our case, this involved dividing \(8\) and \(-4x\) by 2.
• Simplify Expressions: The result should be a clean, straightforward expression with no unnecessary elements, such as \(y = 4 - 2x\).

Simplifying equations not only makes them easier to work with but also provides a clearer understanding of the relationship between variables. This clarity is vital for analyzing or graphing the equations, solving for outputs given specific inputs, and other analytical purposes.