Problem 31 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,-2)\) and \((3,6)\)

Step-by-Step Solution

Answer

The equations of the line, in point-slope form and slope-intercept form, are \(y + 2 = 1.33(x + 3)\) and \(y = 1.33x + 4\) respectively

Step 1: Find the slope of the line

Start by determining the slope of the line, denoted as m, using the slope formula which is \( m = (y_2 - y_1) / (x_2 - x_1) \). Here, \( x_1 = -3, y_1 = -2, x_2 = 3, y_2 = 6 \).

Step 2: Apply slope to point-slope formula

Substitute the slope, m, and one of the points \(-3, -2\) into the point-slope formula, which is \( y - y_1 = m(x - x_1) \). This would give the line equation in point-slope form.

Step 3: Convert into slope-intercept form

Transform the point-slope form into slope-intercept form \(y = mx + c\). This is done by isolating y in the equation.

Key Concepts

Point-Slope Form

When it comes to writing linear equations, the point-slope form is incredibly useful, particularly when you're given a point and the slope of the line. It's written as \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) represents a point on the line and \( m \) indicates the slope of the line.

Consider the problem at hand; we have two points, \((-3,-2)\) and \((3,6)\). By finding the slope, which will be described in detail in the following section, you can easily plug it into this form. Let’s say the calculated slope is \( m \). If we use the point \((-3,-2)\), the point-slope form would be \( y + 2 = m(x + 3) \). This form is incredibly versatile and significantly eases the process of creating linear equations, as it directly incorporates the key characteristics of the line.

Slope-Intercept Form

The slope-intercept form of a linear equation is probably the most familiar one: \( y = mx + b \), where \( m \) is the slope of the line and \( b \) represents the y-intercept, the point where the line crosses the y-axis. Converting from point-slope to slope-intercept form involves rearranging the equation to solve for \( y \).

Using our previous example with the point-slope form \( y + 2 = m(x + 3) \), you would first distribute the slope \( m \) across the \( x + 3 \), then subtract 2 from both sides to isolate \( y \). The result is the linear equation in slope-intercept form.

Example Conversion

If for instance, after the distribution we get \( y + 2 = mx + 3m \), and our slope \( m \) was calculated to be 1, the equation would simplify to \( y = x + 3m - 2 \). If \( 3m \) actually equals 3, our final equation would be \( y = x + 1 \), a neat linear equation showing how each increase in \( x \) increases \( y \) by the same amount.

Slope Calculation

Calculating the slope is a fundamental skill in algebra, as it defines the steepness and direction of a line. The slope is calculated by the formula \( m = (y_2 - y_1) / (x_2 - x_1) \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.

In our exercise, applying the slope formula to the points \((-3,-2)\) and \((3,6)\) gives us \( m = (6 - (-2)) / (3 - (-3)) \), which simplifies to \( m = 8 / 6 \) or \( m = 4 / 3 \). This tells us that for every 3 units we move to the right along the x-axis, the line will rise by 4 units. Remember, a positive slope means the line is ascending from left to right, whereas a negative slope means the line is descending.