Problem 23 - Elementary Algebra

Question

Find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. (Objective 1a) \(m=\frac{3}{5}\) and \(b=2\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{3}{5}x + 2 \).

Step 1: Identify the Formula

We will use the slope-intercept form of a line, which is given by the formula \( y = mx + b \). Here, \( m \) is the slope of the line and \( b \) is the y-intercept.

Step 2: Substitute the Given Values

Substitute the given slope \( m = \frac{3}{5} \) and y-intercept \( b = 2 \) into the slope-intercept formula: \( y = \frac{3}{5}x + 2 \).

Key Concepts

Linear Equations

A linear equation is a mathematical statement that creates a straight line when graphed on a coordinate plane. Every linear equation takes the form of a straight line and is generally written as:

• Standard Form: \( Ax + By = C \)
• Slope-Intercept Form: \( y = mx + b \)

Linear equations are very useful. They describe relationships where there is a constant rate of change between two variables.
Understanding which form to use can help you solve problems more effectively. In our example, we used the slope-intercept form because it provides an easy way to identify and graph the line with a given slope and y-intercept.
Mastering linear equations is essential for success in many areas of mathematics, including algebra, calculus, and beyond.

Slope

The slope of a line is a measure that indicates how steep the line is. It is a key concept in understanding linear equations. When dealing with slope, remember:

• The slope (\( m \)) in the slope-intercept form (\( y = mx + b \)) is the amount by which the \( y \) value changes for every 1 unit change in \( x \).
• If the slope is positive, the line rises from left to right. If it's negative, the line falls from left to right.
• A zero slope means a horizontal line, while an undefined slope indicates a vertical line.

For example, a slope of \( \frac{3}{5} \) means that for every 5 units you move horizontally to the right, the line will move up 3 units.
This ratio helps determine the direction and steepness of the line, making slope a fundamental aspect of graphing equations.

Y-Intercept

The y-intercept is another important component of linear equations, particularly in the slope-intercept form. The y-intercept is:

• The point where the line crosses the y-axis, meaning specifically where \( x = 0 \).
• Represented by \( b \) in the equation \( y = mx + b \).

Knowing the y-intercept helps quickly plot the line on a graph.
For instance, if the y-intercept \( b \) is 2, this means the line will cross the y-axis at the point \( (0,2) \). This is a useful starting point for drawing the line or establishing its position on the plane.
Grasping the concept of the y-intercept provides better clarity and understanding of how a linear equation behaves graphically.