Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. \(y\) -intercept is \(2,\) and (4,-1)

Step-by-Step Solution

Answer

The equation of the line is \(y = -\frac{3}{4}x + 2\).

Step 1: Understanding the Point-Slope Formula

The point-slope formula is used to find the equation of a line given a point \((x_1, y_1)\) on the line and the slope \(m\) of the line. The formula is: \[y - y_1 = m(x - x_1)\] In this case, we are given a point \((4,-1)\) and need to find the slope \(m\).

Step 2: Finding the slope using the y-intercept

We know one point on the line \((4, -1)\) and that the y-intercept \(b\) is \(2\). The y-intercept means the point \((0, 2)\) is also on the line. Use these two points to find the slope \(m\) using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-1)}{0 - 4} = \frac{3}{-4} = -\frac{3}{4}\]

Step 3: Substituting into the Point-Slope Form

Now that we have the slope \(m = -\frac{3}{4}\) and the point \((4, -1)\), substitute these values into the point-slope formula:\[y - (-1) = -\frac{3}{4}(x - 4)\] This simplifies to:\[y + 1 = -\frac{3}{4}x + 3\]

Step 4: Converting to Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\). Solve for \(y\) in equation:\[y + 1 = -\frac{3}{4}x + 3\] Subtract 1 from both sides to isolate \(y\):\[y = -\frac{3}{4}x + 3 - 1\] This simplifies to:\[y = -\frac{3}{4}x + 2\] Now the equation is in slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line equation is a straightforward method for describing linear relationships. It is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) denotes the y-intercept. This form is especially useful because it allows us to see at a glance the rate of change of the line and its position relative to the y-axis.

The value of the y-intercept \( b \) is where the line crosses the y-axis. When you have an equation in slope-intercept form, plotting the line on a graph becomes much simpler. Just start at the y-intercept and use the slope \( m \) to determine the direction and steepness of the line. This method is widely used in algebra because it directly gives two key elements: the starting point on the y-axis, and the rise over run that depicts the slope.

Finding the slope

Finding the slope is a crucial step in writing the equation of a line. The slope, represented by \( m \), indicates how steep the line is and the direction it moves. Calculated as the 'rise over run,' it defines how much the line goes up or down for each step it moves horizontally.

To determine the slope between two points, \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula takes the difference in the y-values and divides it by the difference in the x-values. If the slope is positive, the line ascends as it moves right. If it's negative, the line descends. In the example given, we computed the slope between the points \((4, -1)\) and \((0, 2)\), resulting in \( m = -\frac{3}{4} \), which indicates the line falls as it moves from left to right.

Line Equation

A line equation represents the set of all points on a line, offering a complete description of its path on a graph. Using the point-slope form, \( y - y_1 = m(x - x_1) \), makes it easier to derive the line's equation when a point \((x_1, y_1)\) and the slope \( m \) are known.

For the problem at hand, we used this form to plug in the slope \(-\frac{3}{4}\) and the point \( (4, -1) \). After substituting and simplifying, we derived the equation \( y + 1 = -\frac{3}{4}(x - 4) \). Upon further manipulation, it converts to the slope-intercept form, \( y = -\frac{3}{4}x + 2 \).

This linear equation method not only simplifies finding the equation but also visualizing the line on a coordinate plane. Understanding how to switch between point-slope, slope-intercept, and standard forms of line equations enhances your ability to analyze and interpret linear data.