Problem 75 - A Graphical Approach to College Algebra

Question

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=e^{x+1}\\\&2 x+y=3\end{aligned}$$

Step-by-Step Solution

Answer

Intersection occurs at \( (0.51, 1.98) \).

Step 1: Plot the exponential function

The first equation is \( y = e^{x+1} \). Start by creating a table with different \( x \) values and compute corresponding \( y \) values. For example, if \( x = -1, 0, 1 \), then \( y = e^{0}, e^{1}, e^{2} \) respectively. Use these points to plot the curve on a graph.

Step 2: Plot the linear equation

The second equation is \( 2x + y = 3 \). Rewrite it in slope-intercept form: \( y = -2x + 3 \). Pick some \( x \) values (like \(-1, 0, 1\)) and calculate the corresponding \( y \) values using the equation. Plot these points and draw the line on the same graph as the exponential function.

Step 3: Find the intersection point

On the graph, look where the curve from the exponential function and the line from the linear equation intersect. This point is the solution to the system and where the \( x \) and \( y \) coordinates are the same for both equations.

Step 4: Determine the coordinates of the intersection

Use a graphing tool or manually refine the intersection point to determine the precise coordinates to the nearest hundredth. Checking a table of values or a calculator can help to get high precision.

Step 5: Verify the solution

Substitute the \( x \) and \( y \) coordinates of the intersection point back into both equations \( y = e^{x+1} \) and \( 2x + y = 3 \) to verify that they satisfy both equations.

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in math, depicting processes that grow or decay at an exponential rate. They have the general form of \( y = a \times e^{bx} \), where:

• \( e \) is the base of natural logarithms, approximately 2.718.
• \( a \) is a constant multiplier that vertically stretches or compresses the function.
• \( b \) determines the rate of growth or decay; if \( b > 0 \), the function grows; if \( b < 0 \), it decays.

In the exercise, the exponential function is given by \( y = e^{x+1} \). This string of numbers and letters represents a rapid increase in \( y \) as \( x \) grows. To illustrate graphically, select a few \( x \) values and calculate the corresponding \( y \) values. This process paints a picture of what's happening—a curve that rises and never touches the x-axis. Exponential graphs are always smooth and continuously increasing or decreasing. They help us understand dynamics like population growth, radioactive decay, and more.

Linear Equations

Linear equations form straight lines when graphed. They follow the form \( y = mx + c \), where:

• \( m \) is the slope, describing the steepness or tilt of the line.
• \( c \) is the y-intercept, the point where the line crosses the y-axis.

For the exercise, we look at the linear equation \( 2x + y = 3 \), which is converted to slope-intercept form to become \( y = -2x + 3 \). Here, the slope \(-2\) illustrates that for each unit increase in \( x \), \( y \) decreases by 2. Meanwhile, the y-intercept at 3 means the line crosses the y-axis at that point.
To graph a linear equation, pick a few \( x \) values, calculate the \( y \) outcomes, and plot these on a coordinate plane. Connect the dots, and you have a line. Linear equations highlight concepts of constant speed or cost.

Intersection Points

Finding intersection points of graphs, particularly a line and a curve, involves identifying where their equations hold equal values of \( x \) and \( y \). This graphical collaboration creates a shared solution to both equations simultaneously.

In graphical solutions, after plotting the exponential curve \( y = e^{x+1} \) and the linear line \( y = -2x + 3 \), the intersection point symbolizes where the output of both equations is identical.
To pinpoint this intersection precisely, examine the meeting spot on the graph. Use a graphical tool, zoom in on areas of interest, and refine the xy-coordinates to the nearest hundredth. Finally, always verify the coordinates by plugging them back into both equations to ensure they fit.
Intersection points provide foundational insights into system solutions, showing where conditions align in diverse systems. In real-world contexts, identifying intersections could relate to balancing budgets, matching supply with demand, or syncing clocks.