Problem 99 - Algebra 1

Question

Graph the line that passes through the points. Write its equation in slope- intercept form. (Review \(5.3 \text { for } 5.7)\) $$(6,-2),(-5,-8)$$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = \frac{6}{11}x - \frac{58}{11}\).

Step 1: Finding the slope

First find the slope of the line passing through given points (6,-2) and (-5,-8). The slope \(m\) is given by: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] So, plug the coordinates of the given points: \[m = \frac{-8 - (-2)}{-5 - 6} = \frac{-6}{-11} = \frac{6}{11}\]

Step 2: Writing the point-slope form of the equation

The point-slope form of the line equation is given by: \[y - y_1 = m(x - x_1)\] Substitute the values of one of the points and the slope into the equation: \[y - (-2) = \frac{6}{11}(x - 6)\]

Step 3: Converting to slope-intercept form

The slope-intercept form of the line equation is \(y = mx + b\). So, simplify the equation obtained in step 2 to get it in the slope-intercept form: \[y + 2 = \frac{6}{11}x - \frac{36}{11}\] \[y = \frac{6}{11}x - \frac{36}{11} - 2\] \[y = \frac{6}{11}x - \frac{58}{11}\]

Key Concepts

Graphing Lines

Graphing lines is a fundamental skill in understanding linear equations. To graph a line, one needs at least two points on the line. In the exercise, these points are given as (6, -2) and (-5, -8). Once the coordinates of these points are determined, you can plot them on a coordinate plane.
Begin by locating point (6, -2) on the grid. Start at the origin (0,0), move 6 units to the right and 2 units down. Then, plot the second point (-5, -8). From the origin, move 5 units to the left and 8 units down.

• Mark both points on the grid.
• Use a ruler to draw a straight line connecting these two points.
• This line represents the graph of the linear equation passing through the points.

Graphing helps visualize the relationship between the variables involved and confirms the linear nature of the equation by showing a straight line.

Slope Calculation

The slope calculation is crucial for understanding the steepness and direction of a line. The slope is often denoted by the letter \(m\) and calculated using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \(y_2 - y_1\) represents the change in the y-coordinates, and \(x_2 - x_1\) represents the change in the x-coordinates between two points on the line.
From the original exercise, insert the values from the points (6, -2) and (-5, -8) into the formula to get:

• \(m = \frac{-8 - (-2)}{-5 - 6} = \frac{-6}{-11} = \frac{6}{11}\)

This positive slope tells us the line rises as it moves from left to right. A well-understood slope gives insight into how y changes with x.

Linear Equations

Linear equations are those that create a straight line when graphed. The most common representation of a linear equation is the slope-intercept form: \[y = mx + b\]where \(m\) is the slope and \(b\) is the y-intercept of the line. To find the linear equation of a line through two points, first find the slope, which we have as \(\frac{6}{11}\) from the previous section.
Now, use one of the points, such as (6, -2), to find the y-intercept \(b\). Substitute into the formula derived from the point-slope form, \[y - y_1 = m(x - x_1)\]to convert it into the slope-intercept form:\[y + 2 = \frac{6}{11}x - \frac{36}{11}\]Rearrange to solve for \(y\) and find the intercept:\[y = \frac{6}{11}x - \frac{58}{11}\]This equation represents the line in slope-intercept form. It's valuable because it readily shows both the slope and where the line crosses the y-axis (at \(-\frac{58}{11}\)). This method provides a robust understanding of linear relationships in algebra.