Problem 29 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,-1)\) and \((2,4)\)

Step-by-Step Solution

Answer

The equation of the line in point slope form is \(y = x+2\), and in slope-intercept form is \(y=x+2\).

Step 1: Calculate the Slope

We will first need to calculate the slope using formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points into the formula, the slope thus becomes \(m = \frac{4 -(-1)}{2 -(-3)} = 1\). So the slope of the line is 1.

Step 2: Write the Equation in Point-Slope Form

Let's use point \((-3,-1)\) to write a point-slope equation, using the point-slope form \(y-y_1=m(x-x_1)\), we get \(y -(-1)=1(x-(-3)) \), which simplifies to \(y+1=x+3\). After rearranging we get \(y = x+2\). The equation \(y=x+2\) is in point-slope form.

Step 3: Write the Equation in Slope-Intercept Form

The formula for slope-intercept form is \(y=mx + b\). From the obtained point-slope equation \(y=x+2\), we can see this equation is already in the slope-intercept form, where \(m = 1\), and the y-intercept \(b = 2\) which is the constant. So the equation in slope-intercept form is \(y=x+2\).

Key Concepts

Calculate Slope

Understanding how to calculate the slope of a line is crucial when dealing with equations of a line. The slope, often represented by the letter \(m\), is a measure of how steep a line is. To find the slope between two points, you use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1}\] This formula essentially tells us how much the \(y\)-value changes for a unit change in the \(x\)-value. Let’s break it down:

• \( y_1 \) and \( y_2 \) are the \(y\)-coordinates of your two points.
• \( x_1 \) and \( x_2 \) are the \(x\)-coordinates of your two points.

For instance, given the points \((-3,-1)\) and \((2,4)\), plugging them into the formula gives us: \[ m = \frac{4 - (-1)}{2 - (-3)} = 1 \] This tells us that the slope of the line passing through these points is 1. A slope of 1 means that for every additional unit you move to the right along the \(x\)-axis, you will move up by 1 unit on the \(y\)-axis.

Point-Slope Form

Once you've calculated the slope, writing the equation of a line in the point-slope form becomes straightforward. This form is handy when you have a point and the slope. The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] Here's how you can understand and apply it:

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

In our example, using the slope \( m = 1 \) and the point \((-3,-1)\), the equation becomes: \[ y + 1 = 1(x + 3) \] Simplifying this, we find: \[ y = x + 2 \] This is the line's equation in point-slope form. It's essentially a more visual representation of a line, where a specific point and the overall direction (slope) are highlighted.

Slope-Intercept Form

The slope-intercept form is perhaps the most popular way to express a line equation because it readily reveals both the slope and the \(y\)-intercept. The form is: \[ y = mx + b \] In this format:

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

From our point-slope form example, \( y = x + 2 \), we see this is already in the slope-intercept form, indicating: - The slope \( m \) is 1. - The \(y\)-intercept \( b \) is 2. This form is favored because it gives a quick overview of the line's essential properties. Picking up the equation from this form, you can see how steep the line is and where it will intersect the \(y\)-axis, enhancing your understanding of the line's behavior on the graph.