Problem 6 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=-4,\) passing through \((-5,-2)\)

Step-by-Step Solution

Answer

So, the point-slope form of the linear equation is \(y + 2 = -4(x + 5)\), and this equation in slope-intercept form becomes \(y = -4x -22\).

Step 1: Write the point-slope form of the equation

The point-slope form of a line is given by \(y - y_1 = m(x - x_1)\), where m is the slope and \((x_1, y_1)\) is the point that the line passes through. Substituting \(-4\) for \(m\), \(-5\) for \(x_1\), and \(-2\) for \(y_1\), we get the equation \(y - (-2) = -4(x - (-5))\). Simplifying this gives \(y + 2 = -4(x + 5)\).

Step 2: Convert to slope-intercept form

Slope-intercept form is given by \(y = mx + b\), where m is the slope and b is the y-intercept. To convert our equation to this form, distribute the \(-4\) to get \(y + 2 = -4x - 20\). Then subtract 2 from both sides to isolate \(y\), resulting in the equation \(y = -4x -22\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is a popular way to express the equation of a linear line. It's written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. The y-intercept is the point where the line crosses the vertical y-axis.

Converting the point-slope form to the slope-intercept form can help in quickly visualizing and plotting the line on a graph. It is useful for determining how the line behaves as the x-value changes. For instance, when we use the specific example from the exercise: moving from \(y + 2 = -4(x + 5)\) to \(y = -4x - 22\), you can see the slope \(m = -4\) and y-intercept \(b = -22\). This tells us that the line decreases rapidly, and starts below the origin on the y-axis.

Learning to convert equations into slope-intercept form enhances your understanding and ability to visualize linear relationships easily.

Equation of a Line

An equation of a line depicts the relationship between the x and y coordinates on a graph. It explains how a line behaves and provides all the necessary information for plotting it. Common forms include the point-slope form, the slope-intercept form, and the standard form.

Each form offers different insights.

• The point-slope form \(y - y_1 = m(x - x_1)\) is beneficial when you know a point on the line and the slope, as shown in the exercise.
• The slope-intercept form \(y = mx + b\) highlights the line's slope and y-intercept, making it easy to draw the line on a graph.
• The standard form \(Ax + By = C\) is another representation, helpful in certain algebraic calculations.

To choose the best form, consider what information you have and what you need to find. For the exercise, knowing the point and slope aids in constructing the point-slope form first, which then leads to the slope-intercept form for easy graphing.

Slope of a Line

The slope of a line is a crucial concept in understanding linear equations. It is indicated by the letter \(m\) in both the point-slope and slope-intercept forms and it measures the steepness of the line. The slope is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line.

To calculate the slope, you use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of any two points on the line. This tells us how much the y-value changes for a given change in x. For example, in the exercise, the slope is \(-4\), suggesting that for every unit increase in x, y decreases by four units.

Understanding the slope helps describe the line's direction:

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right, like in our exercise.
• A zero slope denotes a horizontal line, while an undefined slope indicates a vertical line.

Being able to determine and interpret the slope is essential for solving and applying linear equations.