Problem 43 - 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. $$ 9 x+7 y=0 $$

Step-by-Step Solution

Answer

Slope is \(-\frac{9}{7}\), and y-intercept is 0.

Step 1: Rearrange the Equation to Slope-Intercept Form

The standard slope-intercept form of a line is given as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To convert the equation \(9x + 7y = 0\) to this form, we need to solve it for \(y\). Start by subtracting \(9x\) from both sides, obtaining \(7y = -9x\). Then, divide both sides by \(7\) to isolate \(y\), giving us \(y = -\frac{9}{7}x\). This equation shows that \(m = -\frac{9}{7}\) and \(b = 0\).

Step 2: Identify the Slope and Y-Intercept

From the equation in slope-intercept form \(y = -\frac{9}{7}x + 0\), we can directly read off the slope \(m\) as \(-\frac{9}{7}\) and the y-intercept \(b\) as \(0\). So, the slope of the line is \(-\frac{9}{7}\) and the y-intercept is \(0\).

Step 3: Graph the Line Using the Slope and Y-Intercept

To graph the line, start at the y-intercept \(b = 0\), which is the point \( (0, 0) \) on the graph. From this point, use the slope \(-\frac{9}{7}\) to find another point. The slope means that for every increase of 7 units in \(x\), \(y\) decreases by 9 units. Starting from \( (0, 0) \), move 7 units to the right along the x-axis (reaching x = 7) and 9 units down to reach \((7, -9)\). Plot this point and draw a line through \((0,0)\) and \((7, -9)\).

Key Concepts

Graphing Lines

Graphing lines involves creating a visual representation of a linear equation on a coordinate plane. This process helps us to see the relationship between variables. To graph a line, you need at least two points that lie on the line. A reliable method to find these points is using the line's slope and y-intercept, which are derived from rewriting the equation in slope-intercept form.

• Start by plotting the y-intercept on the y-axis. This is your first point.
• From there, use the slope, which is the ratio of the vertical change to the horizontal change between two points, to locate another point on the line.
• With these two points, draw a straight line extending in both directions.

Graphing is not just about finding the points; it helps us understand the equation's behavior, like how steeply or gradually the line inclines or declines. Following these steps ensures accuracy and clarity in constructing your graph.

Linear Equations

Linear equations are mathematical statements that create straight lines when graphed. They are called linear because they represent a straight line. At the core of a linear equation is the general form: \[ ax + by = c \] Where \(a\), \(b\), and \(c\) are constants and \(x\) and \(y\) are variables. The defining characteristic of these equations is their consistency in slope, which means the rate of change between the variables is constant.A linear relationship implies that the line will never curve, and it will maintain the same incline/decline along its entire length. When graphed, the line can move in any direction but always appears straight.
This simplicity makes linear equations foundational in algebra, serving many applications in real-world problems where relationships are consistent and linear.

Finding Slope and Y-Intercept

The easiest way to find the slope and y-intercept of a linear equation is to convert it into the slope-intercept form. This is \[ y = mx + b \], where \(m\) represents the slope and \(b\) is the y-intercept. - **Slope \(m\)**: The slope measures steepness and direction. It is calculated as the "rise over run," meaning the change in \(y\) divided by the change in \(x\). In this context, a positive slope indicates an upward trend, while a negative one denotes a downward trend.- **Y-Intercept \(b\)**: The y-intercept is the point where the line crosses the y-axis. It signifies the value of \(y\) when \(x\) equals zero.To find these values, rearrange the given linear equation into slope-intercept form by isolating \(y\) on one side. This method gives clear insights into both the slope and the y-intercept, allowing for quick graphing and analysis of the line's behavior.