Problem 57 - Contemporary Precalculus

Question

Find an equation for the line satisfying the given conditions. Through (-2,1) with slope 3.

Step-by-Step Solution

Answer

Answer: The equation of the line is y = 3x + 7.

Step 1: Write down the given information

The given point is (-2, 1) and the slope, m, is 3.

Step 2: Use the slope-point form of a linear equation

We will use the slope-point form (y - y1) = m (x - x1) to find the equation of the line.

Step 3: Substitute the point and slope into the formula

Plug in the given point (-2, 1) and the slope, m = 3, into the slope-point form of a linear equation: (y - 1) = 3 (x - (-2))

Step 4: Simplify the equation

Now, we simplify the equation to get the equation of the line: (y - 1) = 3 (x + 2) Now, distribute the 3 to both (x + 2) terms: y - 1 = 3x + 6 Next, add 1 to both sides of the equation to isolate y: y = 3x + 7

Step 5: Write the final equation of the line

Thus, the equation of the line is: y = 3x + 7

Key Concepts

Slope-Point Form

The slope-point form is a powerful tool in linear algebra that helps us quickly find the equation of a line when we know a point on the line and its slope. The formula is written as \[ (y - y_1) = m(x - x_1) \] where

• \( y_1 \) is the y-coordinate of the given point,
• \( x_1 \) is the x-coordinate of the given point,
• \( m \) is the slope of the line.

By substituting the known values into this formula, we can quickly generate the equation of the line. This approach simplifies the process compared to deriving the formula from scratch each time. It's important to correctly input the coordinates and slope into the equation to ensure accuracy.
Once the formula is set up, solving for \( y \) allows us to express the equation in the more familiar slope-intercept form, \( y = mx + b \). This makes it easy to graph or further manipulate the equation as needed.

Slope

The concept of slope is pivotal in understanding linear equations. Slope measures how steep a line is, or how quickly it rises or falls as you move along it horizontally. Mathematically, the slope \( m \) is determined by the ratio of the vertical change (often referred to as the "rise") to the horizontal change (the "run") of a line between two points.In formula terms, the slope \( m \) is expressed as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For a positive slope, the line moves upwards as it goes from left to right. For a negative slope, the line falls as it moves from left to right. Understanding slope is crucial because it characterizes the direction and steepness of a line, which can be universally applied across various mathematical and real-world problems, from graphing lines to predicting trends.In our exercise, the slope is given as 3, which means for every unit increase horizontally, the line rises by three units vertically. This steepness directly influences the final equation and its graphical representation.

Equation of a Line

An equation of a line is a mathematical statement that describes all the points on that line. When we know the slope and a point through which the line passes, we can write this equation.
One common form of the line equation is the slope-intercept form, given by:\[ y = mx + b \]where

• \( m \) is the slope of the line, and
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In solving the exercise, we transformed the slope-point form first into the equation \( (y - 1) = 3(x + 2) \), then simplified to obtain \( y = 3x + 7 \). Here, 3 is the slope \( m \), and 7 is the y-intercept \( b \).It's important to note that different forms of line equations can be used depending on the starting information and the context in which the equation is being used. These include point-slope form, slope-intercept form, and standard form, all of which provide various insights into the behavior and positioning of the line on a graph.