Problem 41 - Elementary Algebra

Question

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$-2 x-11 y=11$$

Step-by-Step Solution

Answer

Slope is \(-\frac{2}{11}\); y-intercept is \(-1\).

Step 1: Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start with the equation \(-2x - 11y = 11\). We need to solve for \(y\) to get it into the form \(y = mx + b\).

Step 2: Solve for y

To isolate \(y\), first move \(-2x\) to the other side of the equation by adding \(2x\) to both sides: \[-11y = 2x + 11\]. Next, divide the entire equation by \(-11\):\[y = -\frac{2}{11}x - 1\].Now, the equation is in the form \(y = mx + b\), where \(m = -\frac{2}{11}\) and \(b = -1\).

Step 3: Identify the Slope and Y-Intercept

From the equation \(y = -\frac{2}{11}x - 1\), we identify:- The slope \(m\) is \(-\frac{2}{11}\).- The y-intercept \(b\) is \(-1\). This means the line crosses the y-axis at \(y = -1\).

Step 4: Graph the Line

Start by plotting the y-intercept on the graph at \((0, -1)\). From this point, use the slope to determine another point on the line. The slope \(-\frac{2}{11}\) indicates that for each step down 2 units in the y-direction, you move 11 units to the right along the x-direction. Plot another point using this slope, and draw a line through these points to create the graph of the equation.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a crucial skill in understanding the nature of lines on a coordinate plane. A linear equation is simply an equation that, when graphed, creates a straight line. The most user-friendly format for graphing is the slope-intercept form: \(y = mx + b\). This form quickly tells you the slope and the y-intercept of the line, making graphing straightforward.

To graph a line given an equation:

• First, convert the equation into the slope-intercept form, \(y = mx + b\). Here, the variable \(m\) is the slope of the line, and \(b\) represents the y-intercept.
• After finding these values, begin by plotting the y-intercept on the graph. This is where the line will cross the y-axis.
• Use the slope to find another point on the line by moving right or left for the x-direction change and up or down for the y-direction change, according to the rise over run indicated by the slope.
• Connect these points with a straight line, extending it in both directions.
• Consider marking arrowheads at the end of the line to show that it continues infinitely.

Practicing this process with different equations can help reinforce these concepts, making graphing linear equations seem more like second nature.

Finding Slope

The slope of a line is a measure of its steepness or the angle of inclination. It tells us how much the line rises or falls as it moves across the x-axis. Often represented by the variable \(m\), the slope can be thought of as 'rise over run,' helping us understand how much the y-value changes with respect to a change in x.To find the slope from an equation in the slope-intercept form, such as \(y = mx + b\):

• Simply identify the coefficient of \(x\). This coefficient is the slope \(m\).

Keep in mind:

• A positive slope means the line rises as it goes from left to right.
• A negative slope means the line falls as it moves from left to right.
• A slope of zero indicates a horizontal line, which has no vertical change.
• An undefined slope, which occurs in equations of the form \(x = a\), represents a vertical line.

Understanding the slope helps in predicting how the line will behave, which is essential for interpreting linear relationships in real-life situations.

Y-Intercept

The y-intercept of a linear equation is simply the point where the line crosses the y-axis. It provides a useful starting position when plotting a line on a graph. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\).When finding the y-intercept from the equation:

• Look for the constant value, \(b\), in the equation after it has been reorganized into \(y = mx + b\).

To graph the line:

• Start by putting a point at \((0, b)\) since this is where the line touches the y-axis.

The y-intercept is not only helpful for graphing but also in real-world applications, as it can indicate starting conditions or initial values in various scenarios, such as the starting point in time-sensitive tasks or budgets without any variables affecting them yet.