Problem 13 - College Algebra

Question

For each of the following exercises, solve the equation for \(y\) in terms of \(x\) . $$2 x=5-3 y$$

Step-by-Step Solution

Answer

The task is to solve for \( y \) in terms of \( x \).

Step 1: Identify the task

We need to solve the given equation for \( y \) in terms of \( x \): \( 2x = 5 - 3y \).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

The task is to solve for \( y \) in terms of \( x \).

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can have one or more variables but no variable in a linear equation is raised to a power other than one.

• For example, in the equation \( 2x = 5 - 3y \), both \( x \) and \( y \) are raised to the first power, making the equation linear.
• Linear equations often represent a straight line when graphed on a coordinate plane.

These equations are widely used because they are simple yet powerful tools for modeling relationships between variables. Whether it's determining the cost of items or understanding speed and distance, linear equations are ubiquitous in both academic and real-world contexts.

Variable Isolation

Variable isolation refers to the process of solving an equation to express one variable solely in terms of other variables, effectively "isolating" it on one side of the equation. It’s like solving a mystery where you try to find the value or expression of a variable from a given equation.
To isolate a particular variable, you'll usually perform operations such as:

• Addition or subtraction to move terms around
• Multiplication or division to adjust coefficients

In the problem \( 2x = 5 - 3y \), we isolate \( y \) with the following steps:- Start by moving all terms involving \( y \) to one side: subtract 5 from both sides to get: \( 2x - 5 = -3y \).- Finally, divide every term by -3 to express \( y \) solely in terms of \( x \): \[ y = \frac{2x - 5}{-3} \]

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to work with. When solving linear equations, this can include using techniques like combining like terms, expanding expressions, and factoring.
For the equation \( 2x = 5 - 3y \), algebraic manipulation helps us transform it into a form where \( y \) is isolated:

• First, identify terms to move: here we want to collect all the \( y \)-related terms together. We subtract 5 from each side.
• Next, handle any coefficients that make isolation difficult by dividing every term by \(-3\), which affects both sides equally.

It is important to keep the equation balanced, maintaining equality by performing identical operations on both sides. These manipulations provide the clarity needed to solve for a variable, making equations appear less complex.