Problem 25 - 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. Passing through \((2,4)\) with \(x\) -intercept \(=-2\)

Step-by-Step Solution

Answer

The point-slope form of the equation of the line is \(y-4 = 1(x-2)\). The slope-intercept form of the equation is \(y = x + 2\).

Step 1: Calculate the slope

Given a point \((x1,y1)\) through which line passes and \(x\)-intercept =\(x2\), since at \(x\)-intercept the value of \(y\) is zero, the slope \(m\) of line can be calculated using the formula: \[m = \frac{y1 - 0}{x1 - x2}\]Substituting the given values we find\[m = \frac{4 - 0}{2 - (-2)} = 1\]

Step 2: Write the point-slope form of the equation

\[y-y1 = m(x-x1)\]Substitute \(m=1\), \(x1=2\), and \(y1=4\) into the formula to get\[y-4 = 1(x-2)\]

Step 3: Convert to slope-intercept form

To convert the point-slope form of the equation to the slope-intercept form, we need to simplify the equation and solve for \(y\). \[y = 1(x-2) + 4\]\[y = x-2+4\]\[y = x + 2\]

Key Concepts

slope-intercept form

The slope-intercept form of a line's equation is one of the most common ways to represent a linear equation. It is given as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
This form is particularly useful because it immediately tells you the slope and y-intercept without additional calculations.

• The slope \( m \) indicates the steepness and direction of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
• The y-intercept \( b \) shows where the line passes through the y-axis, which can be essential for graphing.

Let's consider an example from our problem statement: after calculating the slope of the line and applying it to the point-slope form, you end up with \( y = x + 2 \) as the slope-intercept form.
Here, the slope \( m = 1 \) and the y-intercept \( b = 2 \). This means the line rises one unit for every unit increase in \( x \), and it crosses the y-axis at \( y = 2 \).

equation of a line

Understanding the equation of a line is fundamental in algebra. An equation tells us about all the points that lie on a line.
There are several forms to express this equation, including slope-intercept, point-slope, and standard form.

• Slope-Intercept Form: As discussed, it's \( y = mx + b \).
• Point-Slope Form: It's beneficial when you know the slope and a specific point on the line, given by \( y - y_1 = m(x - x_1) \).
• Standard Form: Often expressed as \( Ax + By = C \), which also relates \( x \) and \( y \) linearly.

In the exercise, you start with understanding a line passing through a known point \((2,4)\) as well as the x-intercept, \(-2\).
By calculating the slope and transferring that information to the point-slope form, we access more straightforward methods like the slope-intercept form for various uses, including graphing or understanding line characteristics.

slope calculation

Calculating the slope of a line is an important step in determining the line's behavior. The slope \( m \) is essentially the ratio of the change in the y-coordinates to the change in the x-coordinates as you move along the line.

• The formula used to calculate the slope between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• For our exercise, knowing the point \((2, 4)\) and the x-intercept \(-2\) where \( y = 0 \), the slope is calculated as \( m = \frac{4 - 0}{2 - (-2)} = 1 \).

Once the slope is identified, it helps in laying the groundwork for using different forms of linear equations, like the point-slope form and further converting it into the slope-intercept form.
This understanding of slope gives insight into how steep the line is and in which direction it moves, pivotal for graphing and predicting values on the line.