Problem 30 - Contemporary Precalculus

Question

Find the equation of the line with slope \(m\) that passes through the given point. $$m=2 ;(-2,1)$$

Step-by-Step Solution

Answer

Answer: The equation of the line is $$y = 2x + 5$$.

Step 1: Identify the given information

We are given the slope \(m = 2\) and the point \((-2, 1)\). We will use these values to find the equation of the line.

Step 2: Use point-slope form

The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\). Substitute the given point \((-2, 1)\) as \((x_1, y_1)\) and the given slope \(m = 2\) into the point-slope equation: \(y - 1 = 2(x - (-2))\)

Step 3: Simplify the equation

Now, simplify the equation to obtain the slope-intercept form: \(y - 1 = 2(x + 2)\) Distribute the slope \(m = 2\) across the parentheses: \(y - 1 = 2x + 4\)

Step 4: Solve for y

Add 1 to both sides of the equation to solve for y: \(y = 2x + 5\)

Step 5: Write the final answer

The equation of the line with slope \(m = 2\) that passes through the point \((-2, 1)\) is: $$y = 2x + 5$$

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a valuable tool for constructing the equation of a line when you are given a point on the line and its slope. This form is written as:
\[ y - y_1 = m(x - x_1) \]
Here:

• \( m \) is the slope of the line, indicating how steep the line is.
• \((x_1, y_1)\) is the point through which the line passes.

This formula essentially states that the difference in the \( y \)-values of any two points on the line is proportional to the difference in their \( x \)-values, the constant of proportionality being the slope \( m \).
When applying this form, you simply substitute the given point and slope into the formula to create a linear equation. It's one of the quickest ways to find the equation of a line when given a single point and the slope.

Slope-Intercept Form

The slope-intercept form is perhaps the most famous and widely used form of linear equations. It provides a neat and easy way to immediately see both the slope and the \( y \)-intercept of a line at a glance. The standard form for this is:
\[ y = mx + b \]
Where:

• \( m \) represents the slope of the line, describing how fast the line rises or falls.
• \( b \) is the \( y \)-intercept, which is the point where the line crosses the \( y \)-axis.

To convert from point-slope form to slope-intercept form, you solve for \( y \) to isolate it on one side of the equation. This step-by-step simplification allows you to express the line in terms of its slope and intercept, giving you a straightforward equation you can easily graph or analyze.
Using this form makes it easier to understand the graphical implications of the line, as you can directly plot the \( y \)-intercept and use the slope to find other points on the line.

Equation of a Line

The equation of a line in the context of linear equations can take several forms, each with its uses depending on the situation. Whether in point-slope, slope-intercept, or standard form, all express the same line.
Finding the equation of a line involves identifying the relationship between its steepness (slope) and where it intersects the axes. The process usually starts by using the point-slope form when certain information is available, like a specific point and a slope.
Once you have the point-slope form, converting to the slope-intercept form is often useful for graphing or conveying simpler information about the line's direction and starting point (the \( y \)-intercept).

• Knowing the slope helps in understanding how much the line rises or falls for each unit increase in \( x \).
• Identifying the \( y \)-intercept lets you know the exact point where the line hits the \( y \)-axis, an important feature for graph plotting.
• We can switch between different forms of equations based on what information is most important for the task at hand.

In essence, the equation of a line captures the entirety of a line's behavior and position on the coordinate plane, and understanding how to manipulate these forms is key to mastering linear algebra.