Problem 16 - The Calculus with Analytic Geometry

Question

$$ \text { Find an equation of the line through the points }(3,-5) \text { and }(1,-2), \text { and put the equation in the slope-intercept form. } $$

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{3}{2}x - \frac{1}{2} \)

Step 1: Identify the coordinates of the points

The given points are (3, -5) and (1, -2). We'll use these points to find the slope of the line and then form the equation of the line.

Step 2: Calculate the slope of the line

The slope formula is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the coordinates (x1, y1) = (3, -5) and (x2, y2) = (1, -2), we get \[ m = \frac{-2 - (-5)}{1 - 3} = \frac{-2 + 5}{-2} = \frac{3}{-2} = -\frac{3}{2} \]

Step 3: Use the slope-intercept form equation

The general form of the equation of a line is \[ y = mx + b \] Using the slope \( m = -\frac{3}{2} \), we substitute one of the points into the equation to find the y-intercept \(b\). Let's use the point (1, -2).

Step 4: Find the y-intercept

Substitute the slope (m) and the point (x, y) into the slope-intercept form: \[ y = mx + b \] \[ -2 = -\frac{3}{2}(1) + b \] Solve for b: \[ -2 = -\frac{3}{2} + b \] Adding \(\frac{3}{2} \) to both sides, we get: \[ b = -2 + \frac{3}{2} = -\frac{4}{2} + \frac{3}{2} = -\frac{1}{2} \]

Step 5: Write the final equation

Now that we have the slope and the y-intercept, we can write the equation of the line in slope-intercept form: \[ y = -\frac{3}{2}x - \frac{1}{2} \]

Key Concepts

slope-intercept form

The slope-intercept form is a way to express the equation of a line. It is easy to use because it shows the slope and the y-intercept directly. The general formula is given by \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept. This form is particularly useful because it allows us to quickly identify and graph the line based on the values of \(m\) and \(b\). For example, in the final equation \( y = -\frac{3}{2}x - \frac{1}{2} \), -\frac{3}{2} is the slope and -\frac{1}{2} is the y-intercept. The graph of this line would start at -\frac{1}{2} on the y-axis and fall by -\frac{3}{2} for every 1 unit increase in x.

coordinate geometry

Coordinate geometry is a branch of geometry where points, lines, and shapes are represented using coordinates. Each point in a plane is identified by a pair of numbers called coordinates. The x-coordinate shows how far to move horizontally from the origin, and the y-coordinate shows how far to move vertically. In our exercise, the given points (3, -5) and (1, -2) are coordinates on the plane. By using these coordinates, we can find the slope and equation of the line passing through them.

slope calculation

The slope of a line measures its steepness and direction. It is calculated using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula subtracts the y-values and x-values of two points on the line and divides them. In our example, using points (3, -5) and (1, -2), the slope calculation is \[ m = \frac{-2 - (-5)}{1 - 3} = \frac{-2 + 5}{-2} = \frac{3}{-2} = -\frac{3}{2} \]. The negative slope indicates that the line is decreasing as it moves from left to right.

point-slope form

The point-slope form is another way to write the equation of a line using a point on the line and the slope. Its general formula is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \( (x_1, y_1)\) is a specific point on the line. This form can be very handy when you have a point and a calculated slope and need to find the equation of the line. For example, using our slope \(m = -\frac{3}{2}\) and point (3, -5), we get \[ y + 5 = -\frac{3}{2}(x - 3) \]. This can then be rearranged into slope-intercept form if needed.