Problem 36 - College Algebra

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 slope-intercept formula: \(y=m x+b\)

Step-by-Step Solution

Answer

The formula for \(m\) is \(m = \frac{y - b}{x}\).

Step 1: Rearrange the Equation

To solve for the variable \(m\) in the slope-intercept formula \(y = mx + b\), we first need to isolate \(m\). Start by subtracting \(b\) from both sides of the equation.\[ y - b = mx \]

Step 2: Solve for m

Now, to isolate \(m\), divide both sides of the equation by \(x\). This step gives us the formula for \(m\).\[ m = \frac{y - b}{x} \]

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write linear equations. It is very popular because it makes it easy to spot the slope and the y-intercept of a line. This form is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. The x variable represents a point on the x-axis, and y the corresponding point on the y-axis.

• The slope \(m\) indicates how steep the line is. It shows how much the y value increases or decreases as the x value increases by 1 unit.
• The y-intercept \(b\) is the starting point of the line when \(x = 0\).

Knowing the slope-intercept form makes drawing graphs and understanding the characteristics of linear equations straightforward.

Rearranging Formulas

Formulas are expressions that relate different variables, and sometimes it's necessary to rearrange them to solve for a specific variable. This means rewriting the equation so the targeted variable stands alone on one side of the equation. Rearranging is crucial when you need to understand how one specific variable is affected by changes in another. For the equation \(y = mx + b\), if we're solving for \(m\), rearrangement helps us see how \(m\) relates to \(y\), \(x\), and \(b\).

• Start by moving other terms to the opposite side of the equation using inverse operations. In the given exercise, subtract \(b\) from both sides to isolate terms with \(m\) (\(y - b = mx\)).
• Then, isolate the desired variable completely. In our case, divide both sides by \(x\) to solve for \(m\) (\(m = \frac{y - b}{x}\)).

Rearranging formulas is a powerful tool that allows you to tailor solutions to specific scenarios or further manipulate equations for analysis.

Isolation of Variables

Isolation of variables is a method used when solving equations to find a specific variable's value. The goal is to have the variable you're solving for on one side of the equation while all the other terms are on the opposite side. This makes it easier to see the direct relationship between the variables. When we isolate a variable, we're effectively expressing it in terms of other variables in the equation.

• To isolate a variable, both sides of the equation should undergo the same operation, which ensures the equation remains balanced.
• In the example, to isolate \(m\), we first subtracted \(b\) (making \(y - b = mx\)) and then divided the whole equation by \(x\) (resulting in \(m = \frac{y - b}{x}\)).

Understanding how to isolate variables is a fundamental skill in algebra, supporting a wide range of problem-solving tasks and real-world applications. It clarifies the independence of the variable of interest, showing its dependence only on certain other variables.