Problem 20 - Algebra for College Students

Question

Solve each formula for the specified variable. \(y=m x+b\) (slope-intercept form of a linear equation) (a) for \(x\) (b) for \(m\)

Step-by-Step Solution

Answer

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

Step 1 - Rearrange for x

Starting with the equation in slope-intercept form: y = mx + bSubtract b from both sides to isolate the term with x:y - b = mxNext, divide both sides by m:\[x = \frac{y - b}{m}\]

Step 2 - Solution for x

The equation solved for x is:\[x = \frac{y - b}{m}\]

Step 3 - Rearrange for m

Starting again from the equation: y = mx + bSubtract b from both sides to isolate the term with m:y - b = mxNext, divide both sides by x:\[m = \frac{y - b}{x}\]

Step 4 - Solution for m

The equation solved for m is:\[m = \frac{y - b}{x}\]

Key Concepts

slope-intercept form

In linear equations, the slope-intercept form is one of the most common ways to represent a straight line.
This form is written as: y = mx + bHere,

• y is the dependent variable
• x is the independent variable
• m is the slope, which indicates the steepness of the line
• b is the y-intercept, which is the point where the line crosses the y-axis.

The slope-intercept form is very helpful for quickly sketching a graph of the equation, as it directly gives you the slope and the y-intercept.

rearranging equations

Rearranging equations is a key skill in algebra.
It involves manipulating the equation to solve for a specific variable.
For example, to solve the equation y = mx + b for x, we need to isolate x:

• Start by subtracting b from both sides: y - b = mx
• Next, divide both sides by m: \[x = \frac{y - b}{m}\]

Now, x is isolated and the equation is solved for x.
Applying similar steps can help rearrange equations for any variable.
This skill is fundamental in solving more complex algebraic problems.

solving for a variable

Solving for a variable means isolating that variable on one side of the equation to find its value.
Consider the equation y = mx + b and solving for m:

• Subtract b from both sides to isolate terms with m: y - b = mx
• Divide by x to solve for m: \[m = \frac{y - b}{x}\]

This process involves performing inverse operations to both sides of the equation to isolate the desired variable.
Solving for a variable is crucial for understanding relationships in equations and finding unknown values.

linear equations

Linear equations represent straight lines and have the general form y = mx + b.
They are called 'linear' because their graph forms a straight line.
Key concepts of linear equations include:

• The slope (m), indicating the rise over run or steepness of the line.
• The y-intercept (b), where the line crosses the y-axis.
• The relationship between the variables x and y, showing how a change in x affects y.

Linear equations are foundational in algebra and appear frequently in various math problems.
Mastering them is essential for progressing to more advanced topics.