Problem 72 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ m=-1, \quad(2,0) $$

Step-by-Step Solution

Answer

Question: Find the equation of the line with a slope of -1 that passes through the point (2, 0), and write the equation in slope-intercept form. Answer: The equation of the line in slope-intercept form is $$y = -1x + 2$$.

Step 1: Write the general equation of a line

The general equation of a line is given by $$y = mx + b$$ where m is the slope and b is the y-intercept.

Step 2: Substitute the given slope

We have the slope \(m = -1\). So we can rewrite the equation by substituting the slope: $$y = -1x + b$$

Step 3: Use the given point to find the y-intercept

Next, we have to find the value of b, the y-intercept. We have a point on the line, \((2,0)\). Substitute the coordinates of the point into the equation: $$0 = -1(2) + b$$ Now, solve for b: $$b = 0 + 2 = 2$$

Step 4: Write the final equation in slope-intercept form

Finally, substitute the value of b back into the equation to get the equation of the line in slope-intercept form: $$y = -1x + 2$$

Key Concepts

Linear Equations

Linear equations are foundational in algebra and describe a straight line on a graph. They have a simple form called the slope-intercept form, expressed as \(y = mx + b\). Here:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) is the slope of the line.
• \(b\) is the y-intercept.

This equation allows you to easily graph a line or understand how it behaves. Every linear equation corresponds to a straight line, making them predictable and straightforward. By using slope-intercept form, we can quickly identify how the line slopes and where it crosses the y-axis. This is particularly helpful in understanding relationships between variables and making predictions.

Slope

The slope \(m\) of a line is a measure of its steepness and direction. In the slope-intercept form \(y = mx + b\), the slope \(m\) tells you how much the y-value changes for each unit increase in the x-value. It can be calculated using two points on the line: \( (x_1, y_1) \) and \( (x_2, y_2) \).The formula for the slope is:\[ \text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1} \]

• If the slope \(m\) is positive, the line rises from left to right.
• If \(m\) is negative, the line falls from left to right.
• A steeper slope indicates a steeper line.
• A zero slope means the line is horizontal.

In the example given, the slope \(m = -1\) indicates a line falling at a consistent rate as x increases.

Y-Intercept

The y-intercept \(b\) is the point where the line crosses the y-axis. This is the value of \(y\) when \(x = 0\). Represented in the slope-intercept form \(y = mx + b\), it is easy to find and interpret:

• The y-intercept helps determine the starting point of the line on the y-axis.
• If \(b\) is positive, the line crosses the y-axis above the origin.
• If \(b\) is negative, it crosses below the origin.
• A y-intercept of zero means the line crosses at the origin.

In our exercise, substituting the point \((2, 0)\) into the equation allowed us to solve for \(b\), finding that the y-intercept is \(b = 2\). This tells us that the line crosses the y-axis at the point \((0, 2)\), providing a clear starting point for graphing.