Type The Correct Answer In The Box. Use Numerals Instead Of Words.Consider This Equation: 10 X + 25 3 Z + 12 = 5 X Z + 4 \frac{10x+25}{3z+12}=\frac{5x}{z+4} 3 Z + 12 10 X + 25 = Z + 4 5 X How Many Valid Solutions Does The Equation Have?The Equation Has □ \square □ Valid Solution(s).
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Introduction
Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of the underlying principles. In this article, we will delve into the world of rational equations and explore a specific equation that will help us understand the concept of valid solutions.
Understanding Rational Equations
A rational equation is an equation in which the unknown variable appears in the numerator or denominator of a fraction. Rational equations can be solved using various techniques, including cross-multiplication, factoring, and the quadratic formula.
Cross-Multiplication Method
The cross-multiplication method is a popular technique used to solve rational equations. This method involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
Factoring Method
The factoring method is another technique used to solve rational equations. This method involves factoring the numerator and denominator of the equation and then canceling out any common factors.
Quadratic Formula Method
The quadratic formula method is a powerful technique used to solve rational equations. This method involves using the quadratic formula to solve for the unknown variable.
Solving the Given Equation
Now that we have a good understanding of rational equations, let's move on to solving the given equation.
Given Equation
Consider the equation:
Step 1: Cross-Multiplication
To solve this equation, we will use the cross-multiplication method. We will multiply both sides of the equation by the LCM of the denominators, which is .
from sympy import symbols, Eq, solve
x, z = symbols('x z')
eq = Eq((10x+25)/(3z+12), (5*x)/(z+4))
eq = Eq((10x+25)(z+4), (5x)(3*z+12))
Step 2: Expand and Simplify
Next, we will expand and simplify the equation.
# Expand and simplify
eq = Eq(10*x*z + 40*x + 25*z + 100, 15*x*z + 60*x)
Step 3: Move All Terms to One Side
Now, we will move all terms to one side of the equation.
# Move all terms to one side
eq = Eq(10*x*z + 40*x + 25*z + 100 - 15*x*z - 60*x, 0)
Step 4: Combine Like Terms
Next, we will combine like terms.
# Combine like terms
eq = Eq(-5*x*z + 40*x + 25*z + 100, 0)
Step 5: Factor Out Common Terms
Now, we will factor out common terms.
# Factor out common terms
eq = Eq(-5*x*(z - 8) + 25*(z + 4), 0)
Step 6: Solve for x
Finally, we will solve for x.
# Solve for x
solution = solve(eq, x)
Conclusion
In this article, we have solved a rational equation using the cross-multiplication method. We have also explored the concept of valid solutions and how to determine the number of valid solutions for a given equation.
Valid Solutions
The given equation has 1 valid solution.
Discussion
The concept of valid solutions is crucial in mathematics, as it helps us understand the behavior of equations and functions. In this article, we have seen how to determine the number of valid solutions for a given equation.
Real-World Applications
Rational equations have numerous real-world applications, including physics, engineering, and economics. In physics, rational equations are used to model the motion of objects, while in engineering, they are used to design and optimize systems. In economics, rational equations are used to model the behavior of markets and economies.
Final Thoughts
In conclusion, rational equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. By using the cross-multiplication method, we can solve rational equations and determine the number of valid solutions.
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Introduction
In our previous article, we explored the concept of rational equations and solved a specific equation using the cross-multiplication method. In this article, we will answer some frequently asked questions about rational equations.
Q: What is a rational equation?
A rational equation is an equation in which the unknown variable appears in the numerator or denominator of a fraction.
Q: How do I solve a rational equation?
There are several methods to solve rational equations, including cross-multiplication, factoring, and the quadratic formula.
Q: What is the cross-multiplication method?
The cross-multiplication method involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
Q: What is the factoring method?
The factoring method involves factoring the numerator and denominator of the equation and then canceling out any common factors.
Q: What is the quadratic formula method?
The quadratic formula method involves using the quadratic formula to solve for the unknown variable.
Q: How do I determine the number of valid solutions for a rational equation?
To determine the number of valid solutions for a rational equation, we need to consider the restrictions on the variable and the behavior of the equation.
Q: What are some real-world applications of rational equations?
Rational equations have numerous real-world applications, including physics, engineering, and economics.
Q: Can you provide an example of a rational equation in physics?
Yes, here is an example of a rational equation in physics:
Consider the equation:
This equation represents the relationship between velocity, time, distance, and speed in a physical system.
Q: Can you provide an example of a rational equation in engineering?
Yes, here is an example of a rational equation in engineering:
Consider the equation:
This equation represents the relationship between current, resistance, voltage, and energy in an electrical circuit.
Q: Can you provide an example of a rational equation in economics?
Yes, here is an example of a rational equation in economics:
Consider the equation:
This equation represents the relationship between price, quantity, revenue, and sales in a market economy.
Q: How do I know if a rational equation has a valid solution?
To determine if a rational equation has a valid solution, we need to consider the restrictions on the variable and the behavior of the equation.
Q: What are some common mistakes to avoid when solving rational equations?
Some common mistakes to avoid when solving rational equations include:
- Not considering the restrictions on the variable
- Not simplifying the equation
- Not checking for extraneous solutions
Conclusion
In this article, we have answered some frequently asked questions about rational equations. We have also explored some real-world applications of rational equations and provided examples of rational equations in physics, engineering, and economics.
Final Thoughts
Rational equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. By using the cross-multiplication method, factoring method, and quadratic formula method, we can solve rational equations and determine the number of valid solutions.