Problem 44 - Elementary Algebra

Question

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$-5 x-13 y=26$$

Step-by-Step Solution

Answer

Slope: -5/13; y-intercept: -2.

Step 1: Convert equation to slope-intercept form

The first step is to convert the given line equation into the slope-intercept form, which is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Start with the equation \(-5x - 13y = 26\).

Step 2: Solve for y

To isolate \(y\), add \(5x\) to both sides of the equation to get \(-13y = 5x + 26\). Then, divide every term by \(-13\) to solve for \(y\): \(y = -\frac{5}{13}x - 2\).

Step 3: Identify the slope and y-intercept

From the equation \(y = -\frac{5}{13}x - 2\), identify the slope \(m = -\frac{5}{13}\) and the y-intercept \(c = -2\).

Step 4: Graph the equation

To graph the line, start by plotting the y-intercept \((0, -2)\) on the graph. Since the slope is \(-\frac{5}{13}\), for every 13 units moved horizontally to the right, move 5 units down vertically. Connect these points to form the line.

Key Concepts

Equation of a Line

An equation of a line is a mathematical expression that describes all the points on that line. It can be represented in several formats, one of the most common being the slope-intercept form, which is written as \( y = mx + c \). In this form:

• \( y \) represents the dependent variable or the y-coordinate crucial for graphing the line.
• \( m \) is the slope of the line, which indicates the line's steepness and direction.
• \( c \) is the y-intercept, the point where the line crosses the y-axis.

Transforming an equation into this form helps us quickly identify these key characteristics that define the line on a graph. By converting the original equation \(-5x - 13y = 26\) into slope-intercept form, we can easily understand how the line behaves on the coordinate plane.

Graphing Linear Equations

Graphing linear equations is a visual representation of all solutions of the equation on a coordinate plane. To graph the equation of a line, first convert it into the slope-intercept form \( y = mx + c \). This form provides the slope and y-intercept directly, which are essential for graphing.

• Begin by plotting the y-intercept \((0, c)\) on the y-axis. For our example, this point is \((0, -2)\).
• Next, use the slope \( m \). The slope \( m \) is a ratio of the vertical change (rise) to the horizontal change (run). From the y-intercept, use this ratio to determine another point on the line. In our case, the slope is \(-\frac{5}{13}\), meaning for every 13 units right, you go 5 units down.
• Draw a line through these points to complete the graph. This will visually display the infinite set of solutions for the equation \(-5x - 13y = 26\).

Graphing makes understanding the relationship and behavior of variables easier, enhancing comprehension of linear relationships in math.

Slope and Y-Intercept

The slope and y-intercept are fundamental aspects of a line's equation in slope-intercept form. They offer insights into a line's direction and position on a graph.

Slope (\(m\))

• The slope indicates how steep a line is and its direction. It’s calculated as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line.
• A positive slope means the line ascends as it moves from left to right, while a negative slope indicates it descends.
• In our example, the slope \(-\frac{5}{13}\) means the line goes down 5 units for every 13 units it moves to the right.

Y-Intercept (\(c\))

• The y-intercept is the y-coordinate of the point where the line crosses the y-axis.
• It tells us the output value when the input (x) is zero.
• In the solution, the y-intercept \(-2\) means the line crosses the y-axis at \((0, -2)\).

Understanding these components helps in quickly sketching the line and predicting its behavior on the plane.