Problem 35 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ (-1,5),(4,5) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is \(y = 5\).

Step 1: Find the slope

To find the slope of the line passing through the points (-1, 5) and (4, 5), we'll use the slope formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Plugging the given points into the slope formula, we have: $$ m = \frac{5-5}{4-(-1)} = \frac{0}{5} = 0 $$ So the slope of the line is 0.

Step 2: Find the y-intercept

Since the slope m = 0, the line is horizontal. A horizontal line has the same y-coordinate for all points on the line. In this case, the y-coordinate is 5. Therefore, the line passes through the y-axis at point (0, 5), so its y-intercept (b) is 5.

Step 3: Write the equation of the line in slope-intercept form

Now that we have the slope (m) and the y-intercept (b), we can write the equation of the line in slope-intercept form (y = mx + b). Substituting the values, we get: $$ y = 0x + 5 $$ Simplifying the equation, we have: $$ y = 5 $$ So the equation of the line in slope-intercept form is \(y = 5\).

Key Concepts

Equation of a Line

The equation of a line is a fundamental concept in algebra and coordinate geometry. It allows us to describe the relationship between two variables, typically represented as \(x\) and \(y\). The most common form for expressing this relationship is the slope-intercept form: \(y = mx + b\), where \(m\) represents the slope and \(b\) signifies the y-intercept. This form is particularly useful because it provides direct information about the line's behavior: how it moves across the graph and where it cuts through the y-axis.

To write the equation of a line that passes through two specific points, we need to determine both the slope and the y-intercept. Once identified, these values can be directly inserted into the slope-intercept formula. This method provides an intuitive way to visualize and analyze linear functions, making it easier to solve algebraic problems and draw graphs in mathematical studies.

Slope Calculation

Calculating the slope is the first step in finding the equation of a line. The slope essentially measures the steepness or incline of the line. It is determined by calculating the change in the \(y\) values divided by the change in the \(x\) values between two points on the line, using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This ratio of vertical change (rise) to horizontal change (run) can tell you a lot about the direction of the line.

In a horizontal line, like the one passing through (-1,5) and (4,5), the slope is \(0\) because there is no change in the \(y\) values (both are 5). This means the line is flat, showing that for any change in \(x\), \(y\) remains constant. High positive slopes indicate steep rising lines, while high negative slopes show steep falling lines. The slope is one of the key components used to define the linear equation in slope-intercept form.

Y-Intercept

The y-intercept is a crucial part of the equation of a line, representing the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the \(b\) component specifically highlights this location on the graph. It tells us the value of \(y\) when \(x\) is zero.

For the line that passes through the points (-1, 5) and (4, 5), the y-intercept is \(5\). The y-intercept helps anchor the line on the graph, providing a starting position from which the line extends at a steepness determined by its slope. For horizontal lines, the graph doesn't incline or decline, so the y-intercept is the only point you need to convey the line's position—in this case, a simple equation \(y = 5\). Recognizing the y-intercept is vital for graphing linear equations and understanding their visual representation.