Problem 23 - Elementary Algebra

Question

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=\frac{3}{5} \text { and } b=2 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. This equation shows how \( y \) relates to \( x \).

Step 2: Substitute the Given Values

Substitute the given values into the slope-intercept form. The problem gives \( m = \frac{3}{5} \) and \( b = 2 \). Inserting these into the equation: \[ y = \frac{3}{5}x + 2 \]

Step 3: Finalize the Equation

The resulting equation after substitution represents the line having the specified slope and y-intercept. So the final equation of the line is \( y = \frac{3}{5}x + 2 \).

Key Concepts

Linear Equations

Linear equations are fundamental in algebra, visually representing straight lines on a graph. They model relationships between variables, typically denoted by "x" and "y." A linear equation expresses a straight line, which means that for every increase in "x," "y" will change at a consistent rate.
The general form of a linear equation is given as:

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

Let's focus on the slope-intercept form, which is the most straightforward way to understand linear equations. In this format, "\( m \)" represents how steep the line is, and "\( b \)" represents where the line crosses the y-axis. This makes it easy to calculate the y-value for any x-value, helping solve various problems in algebra quickly. Being able to grasp linear equations is crucial, as they're extensively used in mathematics to derive functions, understand graphical data, and solve real-world problems.

Slope

The slope of a line signifies how slanted or tilted the line is, indicating its steepness. In the slope-intercept form \( y = mx + b \), "\( m \)" symbolizes the slope.
Here, the slope is computed as the ratio of the vertical change (rise) to the horizontal change (run). Mathematically, it's expressed as:

• \( m = \frac{\text{rise}}{\text{run}} \)

The slope, denoted by "m" in our example, is \( \frac{3}{5} \). This means that for every 5 units the line moves horizontally, it rises 3 units vertically.

• A positive slope, like in our equation, ascends from left to right.
• A negative slope would descend as it moves right.

Understanding the slope is vital for analyzing the direction and angle of lines on a graph, forecasting trends, and making connections between variables.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis on a graph. It's crucial in the slope-intercept form of a linear equation \( y = mx + b \), as it provides a starting point for the line.
In the equation, "b" is the y-intercept. For our given example, the y-intercept is "2." This means that the line crosses the y-axis at \( (0, 2) \).

• The y-intercept provides a specific point, without any dependence on "x," where the value of "y" is known.
• It helps plot the line on a graph, providing a reference point to construct the rest of the line.

By knowing the y-intercept, you can easily graph and understand the equation's impact visually, interlinking the slope and intercept to see the full picture of linear relations.