Solving For X When F(x) G(x) With Linear Equations

Jul 14, 2025 by ADMIN 52 views

Solving for x when f(x) = g(x) in Linear Functions

In mathematics, a fundamental problem involves finding the point where two functions have the same value. This often translates to finding the x-value where two functions, f(x) and g(x), intersect. In this comprehensive guide, we will delve into the process of solving for x when f(x) = g(x), using the specific example of two linear functions:

$\begin{cases} f(x) = -\frac{2}{3}x + 5 \\ g(x) = \frac{1}{9}x - 2 \end{cases}$

This article aims to provide a step-by-step approach to solving this type of problem, ensuring a clear understanding of the underlying concepts and techniques. We will explore the algebraic manipulations required to isolate x and determine the point of intersection. This skill is crucial in various mathematical and real-world applications, such as determining break-even points, analyzing supply and demand curves, and optimizing linear models.

Understanding Linear Functions

Before diving into the solution, it's essential to understand the nature of linear functions. A linear function is a function that forms a straight line when graphed. It can be generally represented in the form:

y = mx + b

Where:

y is the dependent variable
x is the independent variable
m is the slope, representing the rate of change of the function
b is the y-intercept, the point where the line crosses the y-axis

In our case, both f(x) and g(x) are linear functions. Let's identify their slopes and y-intercepts:

f(x) = -2/3x + 5: Here, the slope (m) is -2/3, and the y-intercept (b) is 5.
g(x) = 1/9x - 2: Here, the slope (m) is 1/9, and the y-intercept (b) is -2.

The slope indicates the steepness and direction of the line. A negative slope (as in f(x)) means the line slopes downwards from left to right, while a positive slope (as in g(x)) means the line slopes upwards. The y-intercept is the point where the line crosses the vertical axis. Understanding these components is crucial for visualizing the functions and interpreting their intersection.

Setting the Functions Equal

To find the value of x where f(x) = g(x), we need to set the two function expressions equal to each other. This creates an equation that we can solve for x. The equation is:

-2/3x + 5 = 1/9x - 2

This equation represents the point where the two lines intersect on a graph. At this point, the y-values of both functions are the same, hence f(x) = g(x). The next step involves manipulating this equation to isolate x on one side. This is achieved through a series of algebraic operations, ensuring that the equation remains balanced.

Solving the Equation

Now, let's solve the equation step-by-step:

Combine x terms: To get all the x terms on one side, we can add 2/3x to both sides of the equation:

-2/3x + 5 + 2/3x = 1/9x - 2 + 2/3x

This simplifies to:

5 = 1/9x + 2/3x - 2
Find a common denominator: To add the x terms, we need a common denominator. The least common denominator for 9 and 3 is 9. So, we convert 2/3 to 6/9:

5 = 1/9x + 6/9x - 2
Add the x terms: Now we can add the x terms:

5 = 7/9x - 2
Isolate the x term: To isolate the x term, add 2 to both sides of the equation:

5 + 2 = 7/9x - 2 + 2

This simplifies to:

7 = 7/9x
Solve for x: To solve for x, multiply both sides of the equation by the reciprocal of 7/9, which is 9/7:

7 * (9/7) = (7/9x) * (9/7)

This simplifies to:

9 = x

Therefore, the value of x for which f(x) = g(x) is x = 9.

Verifying the Solution

To ensure our solution is correct, we can substitute x = 9 back into both f(x) and g(x) and see if the results are equal:

f(9) = -2/3(9) + 5 = -6 + 5 = -1
g(9) = 1/9(9) - 2 = 1 - 2 = -1

Since f(9) = -1 and g(9) = -1, our solution x = 9 is verified. This means that the two lines intersect at the point (9, -1) on a graph.

Graphical Interpretation

The solution x = 9 represents the x-coordinate of the point where the graphs of f(x) and g(x) intersect. If we were to plot these two lines on a coordinate plane, we would see that they cross each other at the point (9, -1). This graphical representation provides a visual confirmation of our algebraic solution. The intersection point is a crucial concept in linear algebra and has numerous applications in various fields.

Importance of Showing Work

Throughout the solution process, it's crucial to show each step clearly. This not only helps in avoiding errors but also allows for easier verification and understanding of the solution. Showing your work demonstrates a clear thought process and can be beneficial in educational settings or when presenting solutions to others. Each step should logically follow from the previous one, ensuring a clear and concise solution.

Applications in Real-World Scenarios

Finding the intersection of linear functions has numerous applications in real-world scenarios. For instance:

Break-even analysis: In business, the intersection of cost and revenue functions determines the break-even point, where the business starts making a profit.
Supply and demand: In economics, the intersection of supply and demand curves determines the market equilibrium price and quantity.
Mixture problems: In chemistry and other fields, solving systems of linear equations can help determine the quantities of different substances needed to create a specific mixture.
Distance-rate-time problems: In physics, the intersection of distance functions can help determine when two objects will meet.

These are just a few examples, and the ability to solve for the intersection of linear functions is a valuable skill in various disciplines.

Common Mistakes to Avoid

When solving for x in linear equations, several common mistakes can occur. Being aware of these pitfalls can help you avoid them:

Incorrectly combining like terms: Ensure that you only combine terms that have the same variable and exponent.
Forgetting to distribute: When multiplying a constant by an expression in parentheses, make sure to distribute the constant to all terms inside the parentheses.
Incorrectly applying the order of operations: Remember to follow the order of operations (PEMDAS/BODMAS) when simplifying expressions.
Making arithmetic errors: Double-check your calculations, especially when dealing with fractions and negative numbers.
Not verifying the solution: Always substitute your solution back into the original equation to ensure it is correct.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving linear equations.

Conclusion

In this comprehensive guide, we have demonstrated a step-by-step approach to solving for x when f(x) = g(x) for two linear functions. We began by understanding the nature of linear functions, setting the functions equal to each other, and then systematically solving the resulting equation. We verified our solution and discussed the graphical interpretation and real-world applications of this concept. By following these steps and avoiding common mistakes, you can confidently solve similar problems and apply this skill in various contexts. The ability to find the intersection of linear functions is a fundamental skill in mathematics and has far-reaching implications in various fields. Remember to always show your work, verify your solution, and practice regularly to master this essential concept.

By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications involving linear functions. Practice is key, so try solving similar problems to solidify your understanding and build your confidence.