A Complex Algebraic Equation

Oct 13, 2025 by windailys.com 76 views

A Complex Algebraic Equation

The given equation is a complex algebraic equation that involves a variable $r$ in the numerator and denominator of two fractions. The equation is $\frac{r+5}{4} = \frac{11}{r-7}$ . To find the solution set for this equation, we need to isolate the variable $r$ and solve for its possible values.

Step 1: Cross Multiply

The first step in solving this equation is to cross multiply. This involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. This gives us the equation $(r+5)(r-7) = 4 \times 11$ .

import sympy as sp
r = sp.symbols('r')
equation = (r+5)*(r-7) - 44
solutions = sp.solve(equation, r)

Step 2: Expand and Simplify

Next, we need to expand and simplify the left-hand side of the equation. This involves multiplying out the terms in the parentheses and combining like terms. This gives us the equation $r^2 - 2r - 35 = 44$ .

# Expand and simplify the equation
equation = sp.expand(equation)
equation = sp.simplify(equation)

Step 3: Rearrange the Equation

Now, we need to rearrange the equation to get all the terms on one side. This gives us the equation $r^2 - 2r - 79 = 0$ .

# Rearrange the equation
equation = sp.Eq(equation, 0)

Step 4: Solve the Quadratic Equation

The final step is to solve the quadratic equation. This involves using the quadratic formula or factoring the equation. In this case, we can use the quadratic formula to find the solutions.

# Solve the quadratic equation
solutions = sp.solve(equation, r)

Step 5: Simplify the Solutions

Finally, we need to simplify the solutions. This involves combining like terms and eliminating any unnecessary factors.

# Simplify the solutions
solutions = [sp.simplify(solution) for solution in solutions]

The Solution Set

The solution set for the equation $\frac{r+5}{4} = \frac{11}{r-7}$ is $r = -1 \pm 4 \sqrt{5}$ .

Conclusion

In this article, we have shown how to solve a complex algebraic equation using the quadratic formula. We have also demonstrated how to simplify the solutions and eliminate any unnecessary factors. The solution set for the equation $\frac{r+5}{4} = \frac{11}{r-7}$ is $r = -1 \pm 4 \sqrt{5}$ .

Final Answer

The final answer is: $\boxed{r = -1 \pm 4 \sqrt{5}}$

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about solving the equation $\frac{r+5}{4} = \frac{11}{r-7}$ .

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to cross multiply. This involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa.

Q: How do I expand and simplify the left-hand side of the equation?

A: To expand and simplify the left-hand side of the equation, you need to multiply out the terms in the parentheses and combine like terms.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by the equation $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: How do I use the quadratic formula to solve the equation?

A: To use the quadratic formula to solve the equation, you need to identify the values of $a$ , $b$ , and $c$ in the equation. Then, you can plug these values into the quadratic formula and simplify the expression.

Q: What is the solution set for the equation?

A: The solution set for the equation $\frac{r+5}{4} = \frac{11}{r-7}$ is $r = -1 \pm 4 \sqrt{5}$ .

Q: How do I simplify the solutions?

A: To simplify the solutions, you need to combine like terms and eliminate any unnecessary factors.

Q: What is the final answer?

A: The final answer is $r = -1 \pm 4 \sqrt{5}$ .

Common Mistakes to Avoid

When solving the equation $\frac{r+5}{4} = \frac{11}{r-7}$ , there are several common mistakes to avoid.

Mistake 1: Not Cross Multiplying

One of the most common mistakes is not cross multiplying. This can lead to incorrect solutions and a failure to solve the equation.

Mistake 2: Not Expanding and Simplifying

Another common mistake is not expanding and simplifying the left-hand side of the equation. This can lead to incorrect solutions and a failure to solve the equation.

Mistake 3: Not Using the Quadratic Formula

A third common mistake is not using the quadratic formula to solve the equation. This can lead to incorrect solutions and a failure to solve the equation.

Tips and Tricks

When solving the equation $\frac{r+5}{4} = \frac{11}{r-7}$ , there are several tips and tricks to keep in mind.

Tip 1: Use the Quadratic Formula

One of the most important tips is to use the quadratic formula to solve the equation. This will ensure that you get the correct solutions.

Tip 2: Simplify the Solutions

Another important tip is to simplify the solutions. This will ensure that you get the correct solutions and avoid any unnecessary factors.

Tip 3: Check Your Work

A third important tip is to check your work. This will ensure that you have solved the equation correctly and avoid any mistakes.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving the equation $\frac{r+5}{4} = \frac{11}{r-7}$ . We have also discussed common mistakes to avoid and provided tips and tricks to keep in mind. By following these tips and tricks, you can ensure that you solve the equation correctly and get the correct solutions.

Final Answer

The final answer is: $\boxed{r = -1 \pm 4 \sqrt{5}}$