Simplify The Answer Fully.${ \frac{4}{4x+3} = \frac{5}{x-2} }$

Introduction

Rational equations are a fundamental concept in mathematics, and solving them can be a challenging task. In this article, we will focus on simplifying a rational equation, which involves finding the solution to the equation $\frac{4}{4x+3} = \frac{5}{x-2}$. This equation is a classic example of a rational equation, and solving it requires a step-by-step approach.

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 methods, including cross-multiplication, factoring, and the quadratic formula. In this article, we will use the cross-multiplication method to solve the given rational equation.

Cross-Multiplication Method

The cross-multiplication method involves multiplying both sides of the equation by the denominators of the fractions. This method is useful for solving rational equations that have a single fraction on each side. To apply the cross-multiplication method, we need to multiply both sides of the equation by $(4x+3)(x-2)$.

Step 1: Multiply Both Sides by the Denominators

To apply the cross-multiplication method, we need to multiply both sides of the equation by $(4x+3)(x-2)$. This will eliminate the fractions and allow us to solve for the variable.

from sympy import symbols, Eq, solve
x = symbols('x')

equation = Eq(4/(4*x+3), 5/(x-2))

equation = Eq((4/(4x+3))(4x+3)(x-2), (5/(x-2))(4x+3)*(x-2))

equation = Eq(4*(x-2), 5*(4*x+3))

Step 2: Expand and Simplify the Equation

After multiplying both sides of the equation by the denominators, we need to expand and simplify the equation. This will allow us to isolate the variable and solve for its value.

# Expand and simplify the equation
equation = Eq(4*x - 8, 20*x + 15)

Step 3: Isolate the Variable

To solve for the variable, we need to isolate it on one side of the equation. This can be done by subtracting $4x$ from both sides of the equation and then adding $8$ to both sides.

# Isolate the variable
equation = Eq(-8, 16*x + 15)

Step 4: Solve for the Variable

Finally, we can solve for the variable by subtracting $15$ from both sides of the equation and then dividing both sides by $16$.

# Solve for the variable
equation = Eq(-23, 16*x)
equation = Eq(x, -23/16)

Conclusion

In this article, we have simplified a rational equation using the cross-multiplication method. We have shown that the equation $\frac{4}{4x+3} = \frac{5}{x-2}$ can be solved by multiplying both sides by the denominators, expanding and simplifying the equation, isolating the variable, and solving for its value. The solution to the equation is $x = -\frac{23}{16}$.

Final Answer

The final answer is $\boxed{-\frac{23}{16}}$.

Discussion

The rational equation $\frac{4}{4x+3} = \frac{5}{x-2}$ is a classic example of a rational equation that can be solved using the cross-multiplication method. The solution to the equation is $x = -\frac{23}{16}$. This solution can be verified by plugging it back into the original equation.

Related Topics

• Solving Rational Equations
• Cross-Multiplication Method
• Factoring Method
• Quadratic Formula

References

• [1] "Rational Equations" by Math Open Reference
• [2] "Solving Rational Equations" by Khan Academy
• [3] "Cross-Multiplication Method" by Purplemath

Note: The references provided are for informational purposes only and are not a substitute for the original sources.

Introduction

In our previous article, we simplified a rational equation using the cross-multiplication method. We have shown that the equation $\frac{4}{4x+3} = \frac{5}{x-2}$ can be solved by multiplying both sides by the denominators, expanding and simplifying the equation, isolating the variable, and solving for its value. In this article, we will answer some frequently asked questions related to solving rational equations.

Q&A

Q: What is a rational equation?

A: A rational equation is an equation in which the unknown variable appears in the numerator or denominator of a fraction.

Q: What is the cross-multiplication method?

A: The cross-multiplication method is a technique used to solve rational equations. It involves multiplying both sides of the equation by the denominators of the fractions.

Q: How do I know when to use the cross-multiplication method?

A: You should use the cross-multiplication method when you have a rational equation with a single fraction on each side.

Q: Can I use the cross-multiplication method to solve rational equations with multiple fractions?

A: No, the cross-multiplication method is only suitable for solving rational equations with a single fraction on each side. For rational equations with multiple fractions, you should use other methods such as factoring or the quadratic formula.

Q: How do I expand and simplify the equation after multiplying both sides by the denominators?

A: To expand and simplify the equation, you should multiply out the terms and combine like terms.

Q: How do I isolate the variable?

A: To isolate the variable, you should add or subtract the same value to both sides of the equation and then divide both sides by the coefficient of the variable.

Q: What is the final answer to the equation $\frac{4}{4x+3} = \frac{5}{x-2}$?

A: The final answer to the equation $\frac{4}{4x+3} = \frac{5}{x-2}$ is $x = -\frac{23}{16}$.

Q: Can I verify the solution by plugging it back into the original equation?

A: Yes, you can verify the solution by plugging it back into the original equation.

Conclusion

In this article, we have answered some frequently asked questions related to solving rational equations. We have shown that the cross-multiplication method is a useful technique for solving rational equations with a single fraction on each side. We have also provided some tips and tricks for expanding and simplifying the equation, isolating the variable, and verifying the solution.

Final Answer

The final answer is $\boxed{-\frac{23}{16}}$.

Discussion

Solving rational equations can be a challenging task, but with the right techniques and strategies, you can simplify the answer fully. The cross-multiplication method is a useful technique for solving rational equations with a single fraction on each side. We hope that this article has been helpful in answering some of your questions related to solving rational equations.

Related Topics

• Solving Rational Equations
• Cross-Multiplication Method
• Factoring Method
• Quadratic Formula

References

• [1] "Rational Equations" by Math Open Reference
• [2] "Solving Rational Equations" by Khan Academy
• [3] "Cross-Multiplication Method" by Purplemath

Note: The references provided are for informational purposes only and are not a substitute for the original sources.