Problem 12 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Find the slope and \(y\) intercept of the given line and draw the graph. a. \(2 y+3 x=0\) b. \(\frac{x}{3}+\frac{y}{2}=4\)

Step-by-Step Solution

Answer

a: Slope = -3/2, y-intercept = 0. b: Slope = -2/3, y-intercept = 8.

Step 1: Rearrange equation a to slope-intercept form

Equation a is given as: \[ 2y + 3x = 0 \] To find the slope and y-intercept, rearrange the equation into the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. First, subtract \( 3x \) from both sides:\[ 2y = -3x \] Then, divide both sides by 2:\[ y = -\frac{3}{2}x \]

Step 2: Identify slope and y-intercept for equation a

From the rearranged equation \( y = -\frac{3}{2}x \), we identify that: \( m = -\frac{3}{2} \) (slope) and \( b = 0 \) (y-intercept).

Step 3: Graph line for equation a

To graph the line, start at the y-intercept (0,0). From this point, use the slope to find another point. With a slope of \( -\frac{3}{2} \), move down 3 units and right 2 units to reach the point (2, -3). Draw the line through these points.

Step 4: Rearrange equation b to slope-intercept form

Equation b is given as: \[ \frac{x}{3} + \frac{y}{2} = 4 \] To find the slope and y-intercept, rearrange the equation into the form \( y = mx + b \). First, subtract \( \frac{x}{3} \) from both sides:\[ \frac{y}{2} = -\frac{x}{3} + 4 \] Then, multiply both sides by 2:\[ y = -\frac{2}{3}x + 8 \]

Step 5: Identify slope and y-intercept for equation b

From the rearranged equation \( y = -\frac{2}{3}x + 8 \), we identify that: \( m = -\frac{2}{3} \) (slope) and \( b = 8 \) (y-intercept).

Step 6: Graph line for equation b

To graph the line, start at the y-intercept (0,8). From this point, use the slope to find another point. With a slope of \( -\frac{2}{3} \), move down 2 units and right 3 units to reach the point (3,6). Draw the line through these points.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed. They generally have the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. These kinds of equations are pivotal in algebra and pre-calculus, serving as fundamental building blocks to many advanced topics.

One of the most essential forms of linear equations is the slope-intercept form, expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. This format makes it very easy to identify the crucial attributes needed to graph the line.

Slope

The slope \((m)\) of a linear equation describes the steepness and direction of the line. It is calculated as the ratio between the change in \(y\) (rise) and the change in \(x\) (run). Mathematically, it's written as \(m = \frac{\Delta y}{\Delta x}\).

The slope tells us how quickly one variable changes with respect to another. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. For example, the slope of -\(\frac{3}{2}\) for equation (a) tells us that for every 2 units we move right, the line moves down by 3 units.

Understanding the slope is crucial for graphing lines correctly and interpreting the relationship between variables.

Y-Intercept

The \(y\)-intercept \((b)\) is the point where the line crosses the \(y\)-axis. In the slope-intercept form \(y = mx + b\), the \(y\)-intercept is the constant \(b\). This point is crucial because it provides a starting point for graphing the line.

For instance, in equation (b) \(y = -\frac{2}{3}x + 8\), the \(y\)-intercept is 8. This means the line crosses the \(y\)-axis at the point (0,8). You begin graphing here and then use the slope to find other points.

If there is no constant term in the equation after rearranging, like in equation (a), where it becomes \(y = -\frac{3}{2}x\), the \(y\)-intercept is 0. This means the line passes through the origin (0,0).

Graphing Lines

Graphing lines involves plotting points on a coordinate grid based on the equation of the line. To graph a line in slope-intercept form \(y = mx + b\), you follow these steps:

• Start at the \(y\)-intercept \(b\).
• Use the slope \(m\) to determine the direction and steepness of the line.
• Draw a straight line through the points.

For equation (a), \(y = -\frac{3}{2}x\), the \(y\)-intercept is 0. Start at (0,0), then move down 3 units and right 2 units to the point (2,-3). Connect these points with a straight line.

For equation (b), \(y = -\frac{2}{3}x + 8\), the \(y\)-intercept is 8. Start at (0, 8), then move down 2 units and right 3 units to reach (3,6). Draw a line through these points, and you're done!

Accurate graphing helps visualize relationships between variables and solve problems involving linear equations.