Problem 77 - A Graphical Approach to Precalculus with Limits

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

The solution is approximately \((0.31, 2.38)\).

Step 1: Understand the Equations

We need to solve the system of equations \(y=e^{x+1}\) and \(2x+y=3\) by graphically finding the intersection point. The solution is the \((x, y)\) coordinate where these two equations intersect.

Step 2: Graph the Exponential Equation

Graph the equation \(y=e^{x+1}\). This is an exponential curve that gets shifted one unit left as compared to \(y=e^x\). It rises from left to right and passes through the point \((-1, e^0) = (-1, 1)\).

Step 3: Graph the Linear Equation

Graph the equation \(2x + y = 3\). To do this, find the intercepts. Set \(x = 0\), which gives \(y = 3\). Set \(y = 0\), which gives \(x = 1.5\). Draw the line connecting these intercepts.

Step 4: Determine the Intersection Point

Observe the graphs and identify the intersection point of the line and the exponential curve. To solve graphically, you might zoom in to where the two graphs intersect or use graphing software.

Step 5: Verify the Intersection Point

Assume the intersection is computed around \(x = 0.31\) and \(y = 2.38\) using graphing software or manually read from an accurate graph. Substitute back to check the values: \(y = e^{0.31+1} \approx 2.38\) and \(2(0.31) + 2.38 = 3\). These equations validate our intersection point.

Key Concepts

Exponential Functions

Exponential functions are important mathematical functions characterized by a constant base raised to a variable exponent. The general form is \(y = ab^x\), where \(a\) is the initial value, \(b\) is the base, and \(x\) is the exponent. In the case of our exercise, the exponential function is \(y = e^{x+1}\), where \(e\) is Euler's number, approximately equal to 2.718.

• The graph of an exponential function is a curve that grows rapidly, increasing from left to right as the value of \(x\) becomes larger.
• An important property of exponential functions is that they never touch the x-axis; they approach it but keep getting closer without actually touching it.
• The function \(e^{x+1}\) is a transformation of the basic function \(e^x\), involving a shift one unit to the left due to the \(+1\) inside the exponent.

Graphing \(y = e^{x+1}\), we notice the curve starts from a lower y-value and rapidly rises as \(x\) increases. The point \((-1, 1)\) is significant because it helps us to anchor the graph, showing the value of \(y\) when \(x = -1\). This curve, therefore, sets one boundary for solving our system graphically.

Linear Equations

A linear equation represents a straight line in a coordinate system, with the general form \(ax + by = c\). The line \(2x + y = 3\) is straightforward as it combines both \(x\) and \(y\) as variables.

• We use intercepts to graph the line: set \(x = 0\) to find the y-intercept: \(y = 3\), and set \(y = 0\) to find the x-intercept: \(x = 1.5\).
• By plotting these intercepts on a graph and connecting them, we uphold the linear relationship characterized by uniform increments.

Linear equations, by their nature, have consistent slope and direction, ensuring the line continues infinitely in both directions. When solving a linear equation graphically alongside other functions, like an exponential function, intersections reveal solutions to systems based on common coordinates accessible to both functions.

System of Equations

A system of equations involves finding common solutions to two or more equations. In graphical terms, it is where the graphs of the equations intersect, symbolizing shared answers.

• In our system \(\begin{aligned} y = e^{x+1}, \ 2x + y = 3 \end{aligned}\), both equations have a graph line on the same coordinate plane.
• Graphically, we look for intersection points, meaning where the exponential curve reaches the same values as the linear line.
• Simplifying visually, the intersection \((x, y)\) symbolizes the solution, where both equations concurrently hold true.

To accurately graph and solve, we enhance our precision by zooming or using graphing technology, pinpointing the solution as \((x, y) = (0.31, 2.38)\) through visual validation. Checking your work by plugging these back into the original equations is crucial to confirm the solution's accuracy.