Problem 132 - College Algebra

Question

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set Verify this value by direct substitution into the equation. $$ 5^{x}=3 x+4 $$

Step-by-Step Solution

Answer

The solution is the x-coordinate of the intersection point of \(y = 5^{x}\) and \(y = 3x + 4\). This value should be substituted back into the original equation to confirm its accuracy. The exact solution may vary based on precision level of the graphing tool used and manual interpretation of the intersection point.

Step 1: Graphing the functions

Using a graphing utility, plot the functions \(y = 5^{x}\) and \(y = 3x + 4\). These graphs will give a visual representation for where the solution to the equation might be.

Step 2: Finding the Intersection Point

Observe the plotted graphs. The \(x\)-coordinate of the point where these two graphs intersect will give the solution to the equation, as it represents the value of \(x\) that makes both sides of the equation equal.

Step 3: Determining the Solution Set

If the graphing utility has calculated intersection points, use that value. Otherwise, approximate the value by observing where the two graphs intersect. This is the solution to the equation in the form of a set. In other words, the solution set will be {x}. Where x is the x-coordinate of the intersection point

Step 4: Verifying the solution

Substitute the obtained value of \(x\) back into the original equation \(5^{x} = 3x + 4\) and verify both sides evaluate to the same value. This confirms that the solution found graphically is indeed the correct solution to the equation.

Key Concepts

Intersection Points

To solve equations graphically, the intersection points of functions provide key insights. When you plot two functions on the same set of axes, they may intersect at various places. These intersections are crucial because they indicate the values of the variable, usually denoted as \(x\), for which the two functions yield the same result—where they are equal. Thus, finding these points is essential in determining the solutions of equations.
For example, given the equation \(5^{x} = 3x + 4\), plot both \(y = 5^{x}\) and \(y = 3x + 4\) on a graph. The \(x\)-coordinate of their intersection is the answer to the equation, as it's where both equations simultaneously satisfy \(y\).

• Visualize the intersection to understand the relationship.
• Mark these points as they lead to potential solutions.
• Use the \(x\)-coordinate from the intersection to find the solution set.

Finding these points can be greatly aided by graphing utilities, which can pinpoint intersections with precision.

Function Plotting

Function plotting is a valuable tool in visualizing equations. It helps you see the relationship between variables and analyze the behavior of an equation. By graphing each side of an equation, you can easily find potential solutions through visual interpretation.
When plotting functions, make sure that each function is set to a different color or line style to easily differentiate them. For instance, graph \(y = 5^{x}\) with a solid line and \(y = 3x + 4\) with a dashed line. Doing so helps in identifying the intersection, which is vital to solving the equation.

• Choose a viewing window that properly displays the behavior of both functions.
• Use graphing utilities or graphing calculators for accurate results.
• Adjust the scale and zoom to enhance understanding and visibility.

Function plotting not only aids in solving equations but also deepens understanding of how the equations interact.

Solution Verification

Verifying your solution is a critical step in equation solving. After determining a potential solution from the graph, it's important to plug it back into the original equation to ensure that it works. This process confirms that the \(x\)-coordinate of the intersection indeed satisfies the equation.
For the given equation \(5^{x} = 3x + 4\), suppose you find \(x = a\). Substitute \(a\) back into both sides of the equation to check:

• Calculate \(5^{a}\)
• Calculate \(3\cdot a + 4\)
• Compare the results to confirm they are equal.

If both sides of the equation equate, the verification is successful, affirming the solution's correctness. This step ensures accuracy and builds confidence in your graphing results.

Exponential and Linear Equations

Understanding the characteristics of exponential and linear equations helps in solving problems involving them. Linear equations like \(y = 3x + 4\) represent straight lines, where the rate of change is constant. In contrast, exponential equations such as \(y = 5^{x}\) exhibit a variable rate of change, growing rapidly or decaying depending on their base and exponent.

• Linear equations can intersect the x-axis and y-axis at consistent rates.
• Exponential equations can have growth that dramatically increases or decreases.
• Recognizing these distinctions aids in graphing and solving when both equation types are involved.

By understanding these properties, you can better anticipate how their graphs will look and intersect, guiding you to the solution.