Problem 16 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-5,\) passing through (-4,-2)

Step-by-Step Solution

Answer

The line in point-slope form is \(y + 2 = -5(x + 4)\) and the line in slope-intercept form is \(y = -5x - 22\).

Step 1: Determine Point-Slope Form

Utilize the point-slope form for a linear equation, \(y - y_1 = m(x - x_1)\), by substituting the given slope, \(m = -5\), and the coordinates of the point \((-4, -2)\) into the equation. This gives us \(y - (-2) = -5(x - (-4))\).

Step 2: Simplify the Equation

Simplify the equation obtained in Step 1. The equation simplifies to \(y + 2 = -5(x + 4)\). This can then be further simplified to \(y = -5x - 20 - 2\), which leads to \(y = -5x - 22\). The latter is the point-slope form of the line.

Step 3: Determine Slope-Intercept Form

The form obtained in Step 2, \(y = -5x - 22\), is also the slope-intercept form, since this form is \(y = mx + c\), where \(m = -5\) is the slope (which corresponds to the slope given in the problem) and \(c = -22\) is the y-intercept.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a valuable tool for describing a line when you're given its slope and one point on the line. This formula is written as \( y - y_1 = m(x - x_1) \), where:

• \( m \) is the slope of the line
• \( (x_1, y_1) \) is a known point on the line

To use this form, substitute the slope and the coordinates of the point into the equation. For example, if we have a slope of \(-5\) and the line passes through the point \((-4, -2)\), the equation becomes:\(y - (-2) = -5(x - (-4))\)This simplifies to \( y + 2 = -5(x + 4) \). It's a straightforward way to represent the equation of a line when detailed information is documented.
Once you have this form, you can do further manipulations to convert it to other forms like the slope-intercept form. Remember, point-slope form is particularly useful for quickly crafting an equation when you have a specific point and the slope, without needing other detailed information like the y-intercept.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to write the equation of a line. This form is particularly beloved by many students and teachers for its simplicity and the clear information it provides. It is expressed as \( y = mx + c \), where:

• \( m \) is the slope of the line
• \( c \) is the y-intercept, or where the line crosses the y-axis

In the context of our example, after simplifying the point-slope form, we arrived at the equation \( y = -5x - 22 \). This equation is already in the slope-intercept form.
The term \(-5x\) tells us that the slope is \(-5\), meaning for every one unit you move horizontally on the graph, the line moves five units down vertically. The term \(-22\) indicates the line crosses the y-axis at \(-22\). This form makes it easy to graph a line and understand its behavior at a glance.

Equation of a Line

Finding the equation of a line is a fundamental skill in algebra. It helps you describe how two variables relate to each other graphically. Commonly, a line's equation can be represented in different forms including the point-slope form and the slope-intercept form.
Each form offers unique insights into the line's characteristics.

• The point-slope form gives a direct way to express a line using a point and the line's slope. This form is practical for constructing the equation with minimal information.
• The slope-intercept form provides clarity on the slope and where the line crosses the y-axis, making it very intuitive for graphing.

Translating between these forms involves simple algebraic manipulation. For instance, rearranging terms from the point-slope form can naturally lead to the slope-intercept format, as seen in our example: \( y + 2 = -5(x + 4) \) simplifies to \( y = -5x - 22 \).
Understanding these forms allows for flexibility and deeper comprehension of linear relationships, whether in practical problem-solving or theoretical mathematics.