Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(m\) in the slopeintercept formula: \(y=m x+b\)

Step-by-Step Solution

Answer

\(m = \frac{y-b}{x}\)

Step 1: Understand the target

We need to rearrange the given formula, \(y = mx + b\), to solve for the variable \(m\). This means isolating \(m\) on one side of the equation.

Step 2: Rearrange the formula

Start by subtracting \(b\) from both sides of the equation to move \(b\) to the opposite side. This gives us:\[y - b = mx\]

Step 3: Isolate the variable m

To isolate \(m\), divide both sides of the equation by \(x\). This step will get \(m\) by itself:\[m = \frac{y - b}{x}\]

Step 4: Review and simplify

The solution for \(m\) in the formula \(y = mx + b\) is now \(m = \frac{y - b}{x}\). Double-check the steps to ensure the equation is correctly rearranged and simplified.

Key Concepts

Equation Rearrangement

In algebra, rearranging equations is a fundamental skill that allows you to solve for a desired variable within a formula. This involves manipulating the equation using basic algebraic operations such as addition, subtraction, multiplication, and division. The goal is to reconfigure the equation so that you can express one variable in terms of the others.
For example, consider the equation:

• Start with the equation: \(y = mx + b\)
• To rearrange this equation to solve for \(m\), you need to subtract \(b\) from both sides, achieving \(y - b = mx\).
• Then, you'll divide both sides by \(x\) to isolate \(m\), resulting in \(m = \frac{y - b}{x}\).

Rearrangement is essential because it allows you to follow through on specific queries within a problem by focusing on different parts of the equation.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most commonly-used ways to represent linear relationships:
\(y = mx + b\). In this formula, \(m\) represents the slope of the line, \(x\) is the independent variable, \(y\) is the dependent variable, and \(b\) stands for the y-intercept, which is the point at which the line crosses the y-axis.

• The slope \(m\) indicates the steepness of the line: a larger \(m\) means a steeper incline.
• The intercept \(b\) shows where the line crosses the y-axis.

This form is particularly useful for quickly graphing a line or figuring out its slope and y-intercept just by looking at the equation. It is a versatile tool that can be adapted through algebraic manipulation to solve for any unknown within the formula.

Variable Isolation

Variable isolation is the process of solving an equation to find the value of one specific variable. This is often done by systematically moving other terms from one side of the equation to the other, using operations that maintain equality. The goal is to have the variable of interest on one side of the equation by itself.
Consider solving \(y = mx + b\) for \(m\):

• Identify what needs to be isolated: in this case, \(m\).
• Move any other variables or constants away from \(m\): subtract \(b\) from both sides to get \(y - b = mx\).
• Divide both sides by \(x\) to isolate \(m\), leading to \(m = \frac{y - b}{x}\).

By isolating \(m\), you can directly understand its value in different contexts provided by the equation. This technique is crucial for algebra and understanding relationships between different variables in given formulas.