Problem 3 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(3,4), m=\frac{1}{2}$$

Step-by-Step Solution

Answer

The equation of the line that passes through the point (3,4) and has given slope \(\frac{1}{2}\) written in slope-intercept form is \(y = \frac{1}{2}x + \frac{5}{2}\).

Step 1: Insert the given values into the formula

The slope-intercept form is \(y = mx + b\), here \(x = 3, y = 4\) and the slope (\(m\)) is \(\frac{1}{2}\). Insert these values into the equation: \(4 = \frac{1}{2}(3) + b\).

Step 2: Solve for \(b\)

Now we will solve the equation for \(b\), the y-intercept. First, simplify the right side \(4 = \frac{3}{2} + b\), and then subtract \(\frac{3}{2}\) from both sides to isolate \(b\): \(b = 4 - \frac{3}{2} = \frac{5}{2}\).

Step 3: Write the final equation

Now that we have the slope (\(m\)) and the y-intercept (\(b\)), we can write the final equation of the line: \(y = \frac{1}{2}x + \frac{5}{2}\).

Key Concepts

Slope-Intercept Form

Linear equations can be expressed in different ways, but the slope-intercept form is one of the most straightforward and common methods. This form is ideal for quickly identifying the characteristics of a line. It is represented as:

• \( y = mx + b \)

where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept

This format makes it simple to see how the line behaves. By looking at the equation \( y = \frac{1}{2}x + \frac{5}{2} \), for example, we can easily identify both its slope and y-intercept.
The slope indicates how steep the line is, and the y-intercept tells us where the line crosses the y-axis. Together, these values define a unique straight line.

Slope

The slope of a line is a measure of its steepness and direction. It tells us how much the y-value of a line will rise (or drop) as you move along the x-axis. In the context of the slope-intercept form \( y = mx + b \), the slope is denoted by \( m \).
Here are some important points to remember about slope:

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A zero slope means the line is horizontal.
• An undefined slope means the line is vertical.

For the equation in the problem \( y = \frac{1}{2}x + \frac{5}{2} \), the slope is \( \frac{1}{2} \). This indicates that for every 2 units the line moves horizontally, it rises by 1 unit. Understanding slope helps in visualizing how a line plots on the graph.

Y-Intercept

The y-intercept is an important feature of a linear equation, representing the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). It is the value of \( y \) when \( x \) equals zero.
For the given equation \( y = \frac{1}{2}x + \frac{5}{2} \), the y-intercept is \( \frac{5}{2} \). This means that when the line is plotted on a graph, it will intersect the y-axis at the point \((0, \frac{5}{2})\).
The y-intercept provides a useful starting point for drawing the line and offers insight into the initial value of a function, particularly in real-world applications where \( x \) could represent time or some other independent variable.