Problem 15 - Calculus

Question

Find an equation for the line with slope \(m\) through the point \((a, c)\)

Step-by-Step Solution

Answer

The equation is \( y = m(x - a) + c \).

Step 1: Recall the Formula for a Line

The equation for a line with slope \( m \) that passes through a point \( (x_1, y_1) \) is given by the point-slope form: \( y - y_1 = m(x - x_1) \). This formula is useful for deriving the equation of a line when you know the slope and a point that the line passes through.

Step 2: Substitute Given Values

Use the point \((a, c)\) as \((x_1, y_1)\) in the point-slope form. Substitute \( x_1 = a \) and \( y_1 = c \) into the formula: \( y - c = m(x - a) \).

Step 3: Simplify the Equation

The equation \( y - c = m(x - a) \) is already in a simple form known as the point-slope form. However, it can also be expressed in slope-intercept form if needed by solving for \( y \). Distribute \( m \) on the right-hand side: \( y - c = mx - ma \). Then, solve for \( y \) to get: \( y = mx - ma + c \).

Step 4: Final Equation

The equation of the line, after substituting the given point and slope, is \( y = mx - ma + c \). This represents the line with slope \( m \) passing through the point \( (a, c) \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express the equation of a line. This format is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, which indicates how steep the line is, while \( b \) is the y-intercept, the point where the line crosses the y-axis.
The slope is crucial because it tells us how much the y-value changes for each unit increase in x. For example, if \( m = 2 \), then for each step we take to the right on the x-axis, we move two steps up on the y-axis.
The y-intercept \( b \) is equally important, as it marks where the line starts on the y-axis. In practical terms, \( b \) helps set the starting point of your line when drawing it on a graph. If \( b = -3 \), the line will intersect the y-axis at -3. This form is especially useful when you need a quick visualization of a line.

Point-Slope Form

The point-slope form is another way to express the equation of a line. This form is particularly useful when you know the slope of the line and a specific point through which the line passes. The general equation for point-slope form is \( y - y_1 = m(x - x_1) \).
Here, \( m \) stands for the slope, just as in slope-intercept form, while \( x_1 \) and \( y_1 \) are the coordinates of a known point on the line. Using this form makes it easy and straightforward to create the equation of a line as soon as you know a single point and the slope.
For instance, if you know the line passes through the point \((3, 4)\) and has a slope of \(2\), you can plug these numbers into the formula to get: \( y - 4 = 2(x - 3) \). This is particularly beneficial for lines that do not have clear y-intercepts.

Equation of a Line

Constructing the equation of a line can be approached in different ways, depending on the information you have. Generally, there are two popular methods: using the slope-intercept form and using the point-slope form.
To use the slope-intercept form, \( y = mx + b \), you need to know both the slope and the y-intercept of the line. If these are known, it's straightforward to substitute them into the equation.
On the other hand, the point-slope form \( y - y_1 = m(x - x_1) \) requires you to know the slope and a specific point the line passes through. This is often used when these specifics about the y-intercept aren't readily available.
Converting equations from point-slope to slope-intercept form is a common task. By rearranging \( y - y_1 = m(x - x_1) \), solving for \( y \), and simplifying, you can obtain the slope-intercept form. Understanding these forms makes line equations manageable and provides flexibility in solving various problems.