Problem 79 - Elementary Algebra

Question

Solve for the indicated variable. $$ \text { Solve for } y: \quad a x+b y=c $$

Step-by-Step Solution

Answer

The solution for \( y \) is \( y = \frac{c - ax}{b} \).

Step 1: Identify the equation and the target variable

First, identify the equation given in the problem and the variable we need to solve for. The equation is \( ax + by = c \) and we need to solve for the variable \( y \).

Step 2: Isolate the term with the variable 'y'

To solve for \( y \), we need to isolate the term \( by \) on one side of the equation. Start by subtracting \( ax \) from both sides of the equation:\[ ax + by - ax = c - ax \]Simplifying this gives:\[ by = c - ax \]

Step 3: Solve for 'y'

Now, solve for \( y \) by dividing both sides of the equation by \( b \):\[ \frac{by}{b} = \frac{c - ax}{b} \]This simplifies to:\[ y = \frac{c - ax}{b} \]

Key Concepts

Understanding Algebraic Manipulation

Algebraic manipulation is the art of transforming and rearranging equations to uncover the unknown variables. It involves using fundamental math operations like addition, subtraction, multiplication, and division, alongside properties of equality.

Here's what you need to keep in mind:

• Maintain balance: Whatever operation you perform on one side of the equation, you must do to the other side.
• Clear unwanted terms: Strategically add or subtract terms to isolate one variable.
• Factor if necessary: Sometimes, equations might be easier to handle if you factor them first.

Algebraic manipulation serves as the foundation for solving equations. It is crucial for rearranging equations to make them manageable through systematic steps.

Basics of Linear Equations

Linear equations are equations of the first degree. This means their highest power of the variable is one. They form straight lines when graphed on a coordinate plane.

A standard form for a linear equation in two variables is: \[ ax + by = c \]This format showcases two variables, typically x and y, whose coefficients are paired with constants.

• The coefficients \(a\) and \(b\) represent how the variables weight in the equation.
• \(c\) is the constant that sets the value for the sum of linear terms.

Recognizing a linear equation’s structure is fundamental. It helps in predicting results and understanding how changes affect solutions.

Strategy for Isolating Variables

Isolating variables means getting the targeted variable alone on one side of the equation. This process lets you solve linear equations systematically.

Here's a step-by-step guide:

• Identify the variable you need to isolate, like \(y\) in the equation \(ax + by = c\).
• Move all terms not containing the targeted variable to the other side using addition or subtraction. Example: Subtract \(ax\) from both sides to localize \(by\).
• Once \(by\) stands alone, divide through by any coefficients if necessary. Here, dividing both sides by \(b\) would isolate \(y\): \[ y = \frac{c - ax}{b} \]

This approach of isolating variables helps turn a complex equation into a solvable expression that can provide clear solutions.