Problem 75 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (-1,-7) \text { and }(3,-11) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (-1, -7) and (3, -11) is \(y = -x - 6\).

Step 1: Calculate the slope

The slope of a line, denoted as \(m\), is calculated as the change in \(y\) over the change in \(x\). The given points are (-1, -7) and (3, -11). So the change in y (\(Δy\)) is -11 - (-7) = -4 and the change in \(x\) (\(Δx\)) is 3 - (-1) = 4. Thus, \(m = Δy/Δx = -4/4 = -1\).

Step 2: Substitute the slope and one point into the equation to find the y-intercept

Now that we have the slope, \(m = -1\), we can substitute it and one of the points into the equation to solve for the y-intercept, \(b\). Let's use the point (-1, -7): -7 = -1*(-1) + b Solving for \(b\) gives \(b = -6\).

Key Concepts

Slope Calculation

Understanding the calculation of a slope is essential when dealing with linear equations. The slope represents the steepness and direction of a line. To find the slope, or the rate at which the line rises (or falls), you need two points on the line. This is expressed mathematically as the change in the y-values divided by the change in the x-values, known as the rise over run.

For the two points (-1,-7) and (3,-11), we calculate the slope (m) by subtracting the y-coordinate of the first point from the y-coordinate of the second point and doing the same for the x-coordinates to get the changes in y (\(Δy\)) and x (\(Δx\)). So, with \(Δy = -11 - (-7)\) and \(Δx = 3 - (-1)\), we get \(m = Δy/Δx = (-4)/4\), which simplifies to \(m = -1\).

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form equation \(y = mx + b\), the y-intercept is represented by \(b\). Once you have computed the slope, you can determine the y-intercept by choosing one of the given points and plugging in the values, along with the slope, into the slope-intercept form equation.

In our example, we substitute \(m = -1\) and the point (-1,-7) into the equation to find \(b\). We then solve for \(b\):\(-7 = -1\cdot(-1) + b\), which simplifies to \(b = -6\). Hence, the y-intercept is at the point (0, -6), indicating where the line would cross the y-axis.

Linear Equations

Linear equations form the foundation for graphing lines on the coordinate plane. They can be written in various formats, including slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This format makes graphing straightforward because it highlights the rate of change of the line and its initial value on the y-axis.

When we have the slope and y-intercept from the earlier calculations, we construct the equation of the line that passes through the given points. For the points (-1,-7) and (3,-11), we've found the slope to be \(m = -1\) and the y-intercept \(b = -6\), thus the slope-intercept equation of the line is \(y = -1\cdot x - 6\), or more simply \(y = -x - 6\).