Solving Equations using Algebraic and Graphical Methods

When solving equations, there are various methods one can use, including algebraic and graphical approaches. In this article, we will explore how to solve the equation 5^{x-1} 5^x - 5 using both methods. This will not only provide a comprehensive understanding but also demonstrate the flexibility of different mathematical techniques.

Algebraic Method

To start with, let's rewrite the equation in a more manageable form:

5^{x-1} frac{5^x}{5}

Substituting this into the original equation, we have:

frac{5^x}{5} 5^x - 5

Multiplying both sides by 5 to eliminate the fraction, we get:

5^x 55^x - 5

Further simplifying, we can expand and rearrange the equation:

5^x 5^{x 1} - 25

Let us now simplify the expression by setting y 5^x. The equation becomes:

5y - y - 25 0

Simplifying, we get:

4y - 25 0

Now, solving for y gives:

4y 25
y frac{25}{4}

Substituting back for y gives us:

5^x frac{25}{4}

To solve for x, we take the logarithm of both sides:

x log_5left(frac{25}{4}right)

Using the change of base formula, we can express this as:

x frac{log_{10}left(frac{25}{4}right)}{log_{10}5}

This is the exact solution for x. For a numerical approximation, you can use a calculator. The solution to the equation 5^{x-1} 5^x - 5 is:

x log_5left(frac{25}{4}right)

Graphical Method

In addition to the algebraic method, we can also solve the same equation graphically. This method involves plotting two functions and finding their intersection.

The given equation can be written as:

5^{x-1} - 5^x 5 0

We can represent this as the intersection of two functions:

f(x) 5^{x-1}

g(x) 5^x - 5

Plotting these functions on a graph, we can find the x-coordinate of their intersection point. Through this method, we can visually see that the root occurs somewhere between x 1 and x 2.

Alternative Solutions

As demonstrated in the alternative solutions provided, there are multiple ways to approach this equation. Some methods include:

Using Logs

Another approach involves logarithms:

5^x - 5^{x-1} 5

Multiplying both sides by 5 to eliminate the fraction:

5^{x 1} - 5^x 25

Let y 5^x. The equation becomes:

5y - y - 25 0
4y 25
y frac{25}{4}

Substituting back for y gives:

5^x frac{25}{4}
x log_5left(frac{25}{4}right)
x approx 1.1386294361119891

Using Simplified Logarithmic Form

5^x - 5^{x-1} 5

Multiplying both sides by 5 to eliminate the fraction:

5^{x 1} 25
5^x frac{25}{5}
5^x 5

Taking the logarithm on both sides:

x log_5 5
x 1 - log_5 left(frac{5}{6}right)
x approx 0.88672

Conclusion

Both algebraic and graphical methods provide effective ways to solve the equation 5^{x-1} 5^x - 5. The algebraic method offers an exact solution, while the graphical method provides a visual understanding. The key is to choose the method that best suits the problem at hand.

Related Keywords: equation solving, algebraic method, numerical method