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Chapter 2: Problem 85

Solve each formula for the specified variable. $$ y=m x+b \text { for } m $$

Short Answer

Expert verified
The solution for \( m \) is \( \frac{y - b}{x} \).

Step by step solution

01

Identify the formula

The given formula is: \[ y = mx + b \]
02

Isolate the term with the variable

First, we need to isolate the term that contains the variable we are solving for, which is \( m \). To do this, subtract \( b \) from both sides of the formula: \[ y - b = mx \]
03

Solve for the variable

To solve for \( m \), divide both sides of the equation by \( x \): \[ m = \frac{y - b}{x} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
A linear equation is a type of equation where the highest power of the variable is one. It forms a straight line when graphed on a coordinate plane. The general form of a linear equation in one variable is: \[ y = mx + b \] Here, \(y\) and \(x\) are variables, while \(m\) and \(b\) are constants. \(m\) represents the slope, which indicates the steepness of the line, and \(b\) is the y-intercept, where the line crosses the y-axis.
Isolating Variables
Isolating variables is a key step in solving equations. It means getting the variable you are solving for alone on one side of the equation. In our example, we needed to solve for \(m\). To isolate \(m\), we performed the following steps:
  • Subtracted \(b\) from both sides to get rid of the constant term.
  • Divided both sides by \(x\) to isolate \(m\).
The result was: \[ m = \frac{y - b}{x} \] These operations preserve the equality, meaning both sides of the equation remain equal.
Algebraic Manipulation
Algebraic manipulation involves using mathematical operations to rearrange equations. Here are some common operations useful in isolating variables:
  • Addition/Subtraction: Add or subtract the same value from both sides to cancel terms out.
  • Multiplication/Division: Multiply or divide both sides by the same non-zero value to simplify the expression.
  • Combining Like Terms: Group similar terms to simplify equations.
In our example, we subtracted \(b\) and divided by \(x\). These operations are forms of algebraic manipulation that helped us solve for \(m\).
Formula Rearrangement
Rearranging formulas is an essential skill in algebra. It allows you to solve for any variable within an equation. Consider the given formula: \[ y = mx + b \] To solve for \(m\), we rearranged this formula by first isolating the term with \(m\) and then using basic algebraic operations to express \(m\) explicitly:
  • Subtract \(b\) from both sides: \( y - b = mx \)
  • Divide both sides by \(x\): \( m = \frac{y - b}{x} \)
This process of rearranging formulas is useful in many mathematical and real-world problems, enabling you to find unknown values from known quantities.

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