Problem 21 - Elementary Algebra

Question

Write the equation of the line that has slope 4 and passes through the point (-1,2) .

Step-by-Step Solution

Answer

Answer: The equation of the line is y = 4x + 6.

Step 1: Write down the point-slope form equation of the line.

The point-slope form equation of the line is given by: y - y1 = m(x - x1)

Step 2: Plug in the given values.

We are given the slope m=4, and a point (-1, 2). So we substitute these values into the point-slope form equation: y - 2 = 4(x - (-1))

Step 3: Transform the equation to slope-intercept form.

Simplify the equation by performing operations and isolating y on one side of the equation. y - 2 = 4(x + 1) y - 2 = 4x + 4 y = 4x + 6

Step 4: Write the final equation of the line.

The equation of the line in slope-intercept form is: y = 4x + 6

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express the equation of a line in mathematics. This form is characterized by its clear representation of both the slope and the y-intercept of a line.
The general formula for the slope-intercept form is:\[ y = mx + b \]where:

• \( m \) represents the slope of the line, indicating how steep it is.
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

This form is very useful because it allows us to quickly identify the slope and the point where the line crosses the y-axis, making graphing or analyzing linear relationships straightforward.
To convert an equation from another form, such as point-slope, into slope-intercept form, we solve for \( y \) to isolate it on one side of the equation. This method is particularly handy when you need to visualize or graph a linear equation.

Point-Slope Form

The point-slope form is another way of expressing the equation of a line. It is especially useful when you know one point on the line and the slope, but not necessarily the y-intercept. The formula is:
\[ y - y_1 = m(x - x_1) \]where:

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

This form basically describes the line based on the given point and slope. It's a starting point for further manipulation, often used to eventually find the slope-intercept form of the line.
In the given problem, with a slope of 4 and passing through the point \((-1, 2)\), we substitute these values into the point-slope formula to get:
\[ y - 2 = 4(x + 1) \]Here, this initial structure sets us up to further simplify the equation into a more universally used form.

Linear Equations

Linear equations represent straight lines on a graph. They are characterized by their constant rate of change, or slope, across their domain. In essence, each linear equation can be described by its slope and any point it passes through.
The key characteristic of linear equations is their ability to model a direct proportionality between two variables. This makes them applicable in numerous real-world scenarios, such as calculating speed, estimating profit, or predicting trends.
For a linear equation, the slope determines how steep the line is, while the intercepts provide insight into where the line crosses the respective axes.
In practical terms, the knowledge of linear equations allows us to:

• Quickly determine the direction and steepness of a line based on its slope.
• Predict the value of one variable based on another, keeping the relationship linear.

Understanding these foundational elements of linear equations is crucial for solving problems that involve trends and predictions in algebra and beyond.