Problem 44 - Elementary Algebra

Question

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ -5 x-13 y=26 $$

Step-by-Step Solution

Answer

Slope: \(-\frac{5}{13}\), y-intercept: -2.

Step 1: Rewrite the equation in slope-intercept form

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start by isolating \(y\) in the equation \(-5x - 13y = 26\). First, move the \(-5x\) to the right side: \(-13y = 5x + 26\).

Step 2: Solve for y

To isolate \(y\), divide every term by \(-13\), resulting in \(y = -\frac{5}{13}x - 2\). Now, the equation is in slope-intercept form.

Step 3: Identify the slope and y-intercept

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

Step 4: Graph the line

Start by plotting the y-intercept \((0, -2)\) on the graph. Using the slope \(-\frac{5}{13}\), move down 5 units and right 13 units from the y-intercept to find another point. Connect these points with a straight line. Extend the line across the graph.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a vital concept in understanding linear equations. This form is expressed as \(y = mx + b\), where \(m\) stands for the slope and \(b\) is the y-intercept. The beauty of this form is its simplicity in interpreting graphs.

When an equation is expressed in this form, it provides direct insights into two important features of a line:

• Slope (\(m\)): This measure tells us how steep the line is. A positive slope means the line ascends from left to right, while a negative slope means it descends.
• Y-intercept (\(b\)): This is where the line crosses the y-axis. It provides a starting point for graphing the line.

By converting any linear equation into the slope-intercept form, we facilitate an easier path to graphing and analyzing the linear relationship it represents.

Slope and Y-Intercept

Understanding slope and y-intercept is crucial for graphing lines. Let's break it down:- The slope, denoted as \(m\), outlines the tilt of the line. It's calculated as the change in \(y\) divided by the change in \(x\), also known as "rise over run." In the given problem, the slope is \(-\frac{5}{13}\), indicating that for every 13 units you move rightward, the line drops 5 units.- The y-intercept, shown as \(b\) in the equation, is the location on the y-axis where the line crosses. It's essentially the value of \(y\) when \(x = 0\). Here, the y-intercept is \(-2\), suggesting that the line touches the y-axis at the point \((0, -2)\).

Recognizing these components lets you pinpoint a line on a graph effortlessly, serving as a roadmap for sketching the graph accurately.

Graphing Techniques

Graphing a linear equation can be straightforward with the right techniques. Start by using the y-intercept as your anchor point. For the equation \(y = -\frac{5}{13}x - 2\), the y-intercept is \((0, -2)\). Plot this point on the graph, as it marks where the line will cross the y-axis.

Next, incorporate the slope to find another point:

• The slope \(-\frac{5}{13}\) tells us to move down 5 units and over to the right 13 units starting from the y-intercept.
• Locate this second point on the graph.

Finally, draw a straight line through your plotted points, extending it across the graph.

This step-by-step method ensures a precise representation of the linear equation, showcasing the relationship between \(x\) and \(y\) visually.