Problem 40 - Precalculus

Question

In each of parts (a) through (d), first solve the equation for \(y\) so that you can enter it in your graphing utility. Then use the graphing utility to graph the equation in an appropriate viewing rectangle. In each case, the graph is a line. Given that the \(x\) - and \(y\) -intercepts are (in every case here) integers, read their values off the screen and write them down for easy reference when you get to part (e). (a) \(\frac{x}{2}+\frac{y}{3}=1\) (c) \(\frac{x}{6}+\frac{y}{5}=1\) (b) \(\frac{x}{-2}+\frac{y}{-3}=1\) (d) \(\frac{x}{-6}+\frac{y}{-5}=1\) (e) On the basis of your results in parts (a) through (d), describe, in general, the graph of the equation \(\frac{x}{a}+\frac{y}{b}=1,\) where \(a\) and \(b\) are nonzero constants.

Step-by-Step Solution

Answer

The graph is a line with intercepts \((b, 0)\) and \((0, a)\).

Step 1: Solve Equation for y in (a)

Start with the equation \(\frac{x}{2} + \frac{y}{3} = 1\). To isolate \(y\), subtract \(\frac{x}{2}\) from both sides: \[ \frac{y}{3} = 1 - \frac{x}{2} \]Multiply every term by 3 to solve for \(y\): \[ y = 3 - \frac{3x}{2} \]

Step 2: Solve Equation for y in (b)

Begin with \(\frac{x}{-2} + \frac{y}{-3} = 1\). Add \(\frac{x}{2}\) to both sides to isolate \(y\): \[ \frac{y}{-3} = 1 + \frac{x}{2} \]Multiply every term by -3:\[ y = -3 - \frac{3x}{2} \]

Step 3: Solve Equation for y in (c)

Start with \(\frac{x}{6} + \frac{y}{5} = 1\). Subtract \(\frac{x}{6}\) from both sides:\[ \frac{y}{5} = 1 - \frac{x}{6} \]Multiply everything by 5:\[ y = 5 - \frac{5x}{6} \]

Step 4: Solve Equation for y in (d)

Begin with \(\frac{x}{-6} + \frac{y}{-5} = 1\). Add \(\frac{x}{6}\) to both sides:\[ \frac{y}{-5} = 1 + \frac{x}{6} \]Multiply everything by -5:\[ y = -5 - \frac{5x}{6} \]

Step 5: Determine x- and y-Intercepts

Graph the equations to find intercepts: - For (a): Set \(x = 0\), solve for \(y = 3\) and set \(y = 0\), solve for \(x = 2\) gives intercepts (0, 3) and (2, 0).- For (b): Set \(x = 0\), solve for \(y = -3\) and set \(y = 0\), solve for \(x = -2\) gives intercepts (0, -3) and (-2, 0).- For (c): Set \(x = 0\), solve for \(y = 5\) and set \(y = 0\), solve for \(x = 6\) gives intercepts (0, 5) and (6, 0).- For (d): Set \(x = 0\), solve for \(y = -5\) and set \(y = 0\), solve for \(x = -6\) gives intercepts (0, -5) and (-6, 0).

Step 6: Generalize the Graph

For any \(\frac{x}{a} + \frac{y}{b} = 1\), the intercepts are \((b, 0)\) and \((0, a)\). The graph is a straight line, cutting the x-axis at \(b\) and the y-axis at \(a\), with a slope calculated from these intercepts.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help us visualize equations in a simple and efficient way. These can be physical devices like graphing calculators, or software applications found on computers and smartphones. Using these tools, you can input an equation and immediately see its graph on the screen. This provides a visual representation of the mathematical equation and helps in understanding complex concepts like slopes and intercepts. Graphing utilities make it easy to adjust the view of your graph by zooming in or out and altering the range of the axes to get a clearer picture.

With a graphing utility, once you input the equation, such as \(y = 3 - \frac{3x}{2}\), it plots the line on a coordinate grid. You can see where it crosses the x-axis and the y-axis, which corresponds to the intercepts. This visualization assists in understanding the exact behavior of the equation, helping students find intercepts more easily.

X-Intercepts

The x-intercept is the point where a graph crosses the x-axis. At this point, the value of \(y\) is zero. This is important because it shows where the line intersects with the axis that represents zero for \(y\). To find the x-intercept of the equation \(y = 3 - \frac{3x}{2}\), you set \(y\) to 0 and solve for \(x\).

• Example: For the equation \(y = 3 - \frac{3x}{2}\), setting \(y = 0\) gives \[0 = 3 - \frac{3x}{2}\]
• Solving for \(x\) results in \[3 = \frac{3x}{2}\]
• Multiply every term by 2 to find \(x\): \[6 = 3x\]
• Divide by 3: \[x = 2\]

So, the x-intercept is (2, 0). This shows that the line will cross the x-axis at the point (2, 0). Knowing the x-intercept helps in sketching accurate graphs and understanding where a line meets the \(x\) axis.

Y-Intercepts

The y-intercept is the point where the line crosses the y-axis, and at this point, the value of \(x\) is zero. It provides the starting point for the line on the graph and is an essential element for graphing linear equations. To find the y-intercept from the equation \(y = 3 - \frac{3x}{2}\), set \(x = 0\).

• Start from: \[ y = 3 - \frac{3(0)}{2} \]
• Simplifying gives \[y = 3 - 0 \]
• Hence, \[ y = 3 \]

The y-intercept is (0, 3). This implies the line intersects the y-axis at (0, 3), providing a clear reference point from which to draw the rest of the line.

Slope

The slope of a line represents how steep the line is. It is a measure of the line's inclination and is typically described as 'rise over run'. The slope quantifies the rate of change in \(y\) with respect to \(x\). In slope-intercept form, \(y = mx + b\), \(m\) denotes the slope.
Using the equation \(y = 3 - \frac{3x}{2}\), the slope \(m\) is \(-\frac{3}{2}\).

• A negative slope like \(-\frac{3}{2}\) indicates the line slopes downwards from left to right.
• A positive slope indicates it rises from left to right.
• A zero slope denotes a horizontal line.
• An undefined slope implies a vertical line.

In our case, the slope tells us for every 2 units we move right along the x-axis, the line moves 3 units down on the y-axis. Understanding the slope is essential for predicting the direction and steepness of the line.

Coordinates

Coordinates are fundamental to understanding positions on a graph. They denote specific points on the graph in the form \((x, y)\). Each coordinate represents a point where the line either crosses the axis or is plotted along the grid.
Coordinates are used in skills ranging from finding intersections to plotting intercepts, enabling detailed line graphs.

• For example, the coordinate (0, 3) symbolizes the y-intercept on the graph.
• (2, 0) represents the x-intercept on the graph.
• They help in pinpointing exact positions where the line crosses each axis.

Understanding how to read and use coordinates is crucial for graphing equations, solving problems involving distance between points, and determining regions of interest in algebraic contexts.