Problem 8 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((0,2)\) with slope 4

Step-by-Step Solution

Answer

The equation of the line is \(y = 4x + 2\).

Step 1: Identify the point and slope

We are given a point \((0,2)\) and a slope \(m = 4\).

Step 2: Use the point-slope form equation

The point-slope form of a line's equation is \(y - y_1 = m(x - x_1)\). Plug in the point \((x_1, y_1) = (0, 2)\) and slope \(m = 4\). This gives us the equation: \[y - 2 = 4(x - 0)\].

Step 3: Simplify the equation

Simplify \(y - 2 = 4(x - 0)\) to standard slope-intercept form \(y = mx + b\). After simplification, it becomes: \[y - 2 = 4x\] \[y = 4x + 2\].

Key Concepts

Point-Slope Form

The point-slope form is a way to write the equation of a line if you know a point on the line and the slope. It's very useful for quickly finding the equation of a line when you have these ingredients.
The general formula for the point-slope form is \( y - y_1 = m(x - x_1) \)
Here,

• \( (x_1, y_1) \) is a point on the line
• \( m \) is the slope of the line

Using the example in the exercise, the given point is \( (0,2) \) and the slope \( m = 4 \). So plugging these values into the point-slope formula gives us: \( y - 2 = 4(x - 0) \).
This equation represents the line passing through \( (0,2) \) with a slope of 4 in point-slope form.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It's incredibly straightforward to use and interpret.
The general formula for the slope-intercept form is \( y = mx + b \)

• \( m \) represents the slope of the line
• \( b \) is the y-intercept of the line, which is where the line crosses the y-axis

In our example, after we've used the point-slope form equation \( y - 2 = 4(x - 0) \), we simplify it to convert it into the slope-intercept form. First, distribute the 4 on the right side: \( y - 2 = 4x \). Then add 2 to both sides to isolate \( y \): \( y = 4x + 2 \). Now, we have the line in slope-intercept form, where \( m = 4 \) and \( b = 2 \). This tells us that the slope is 4 and the line crosses the y-axis at \( (0,2) \).

Slope of a Line

The slope of a line is a measure of its steepness and direction. It tells you how much the line rises or falls as you move from one point to another.
The slope \( m \) is calculated as the change in y divided by the change in x between two points: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
In simpler terms, the slope is the 'rise' over the 'run'.

• A positive slope means the line is going upwards from left to right
• A negative slope means the line is going downwards from left to right

In our example, the slope given is \( m = 4 \). This means for every unit we move to the right along the x-axis, the line moves 4 units up along the y-axis. Understanding the slope helps in plotting the line and visualizing its orientation on a graph.