Solve The System Of Equations:${ \begin{cases} \frac{2}{x} + \frac{3}{y} = -2 \ \frac{4}{x} - \frac{5}{y} = 1 \end{cases} }$

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Introduction

Solving a system of equations with fractions can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will discuss how to solve a system of equations with fractions using algebraic methods. We will use the given system of equations as an example to illustrate the steps involved in solving it.

The System of Equations

The given system of equations is:

{2x+3y=−24x−5y=1\begin{cases} \frac{2}{x} + \frac{3}{y} = -2 \\ \frac{4}{x} - \frac{5}{y} = 1 \end{cases}

To solve this system of equations, we need to eliminate the fractions and then use algebraic methods to find the values of x and y.

Eliminating the Fractions

To eliminate the fractions, we can multiply both sides of each equation by the least common multiple (LCM) of the denominators. In this case, the LCM of x and y is xy.

Multiplying the first equation by xy, we get:

2y+3x=−2xy2y + 3x = -2xy

Multiplying the second equation by xy, we get:

4y−5x=xy4y - 5x = xy

Solving the System of Equations

Now that we have eliminated the fractions, we can use algebraic methods to solve the system of equations. We can start by adding the two equations to eliminate the x term.

Adding the two equations, we get:

(2y+3x)+(4y−5x)=−2xy+xy(2y + 3x) + (4y - 5x) = -2xy + xy

Simplifying the equation, we get:

6y−2x=−xy6y - 2x = -xy

Solving for x

We can solve for x by isolating it on one side of the equation. We can do this by adding xy to both sides of the equation and then dividing both sides by -2.

Adding xy to both sides, we get:

6y−2x+xy=−xy+xy6y - 2x + xy = -xy + xy

Simplifying the equation, we get:

6y−2x+xy=06y - 2x + xy = 0

Dividing both sides by -2, we get:

−3y+x+12xy=0-3y + x + \frac{1}{2}xy = 0

Solving for y

We can solve for y by isolating it on one side of the equation. We can do this by subtracting xy from both sides of the equation and then dividing both sides by 6.

Subtracting xy from both sides, we get:

−3y+x=−xy-3y + x = -xy

Dividing both sides by 6, we get:

−12y+16x=−16xy-\frac{1}{2}y + \frac{1}{6}x = -\frac{1}{6}xy

Finding the Values of x and y

Now that we have solved for x and y, we can find the values of x and y by substituting the values of x and y into one of the original equations.

Substituting the value of x into the first equation, we get:

2x+3y=−2\frac{2}{x} + \frac{3}{y} = -2

Substituting the value of y into the equation, we get:

2x+312x=−2\frac{2}{x} + \frac{3}{\frac{1}{2}x} = -2

Simplifying the equation, we get:

2x+6x=−2\frac{2}{x} + \frac{6}{x} = -2

Combining the fractions, we get:

8x=−2\frac{8}{x} = -2

Multiplying both sides by x, we get:

8=−2x8 = -2x

Dividing both sides by -2, we get:

x=−4x = -4

Conclusion

In this article, we discussed how to solve a system of equations with fractions using algebraic methods. We used the given system of equations as an example to illustrate the steps involved in solving it. We eliminated the fractions by multiplying both sides of each equation by the least common multiple of the denominators and then used algebraic methods to find the values of x and y. We found the values of x and y by substituting the values of x and y into one of the original equations.

Final Answer

The final answer is x = -4 and y = 1/2.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

  1. Multiply both sides of each equation by the least common multiple of the denominators.
  2. Add the two equations to eliminate the x term.
  3. Simplify the equation and solve for x.
  4. Solve for y by isolating it on one side of the equation.
  5. Find the values of x and y by substituting the values of x and y into one of the original equations.

Tips and Tricks

Here are some tips and tricks to help you solve systems of equations with fractions:

  • Multiply both sides of each equation by the least common multiple of the denominators to eliminate the fractions.
  • Use algebraic methods to solve the system of equations.
  • Add the two equations to eliminate the x term.
  • Simplify the equation and solve for x.
  • Solve for y by isolating it on one side of the equation.
  • Find the values of x and y by substituting the values of x and y into one of the original equations.

Common Mistakes

Here are some common mistakes to avoid when solving systems of equations with fractions:

  • Not multiplying both sides of each equation by the least common multiple of the denominators.
  • Not using algebraic methods to solve the system of equations.
  • Not adding the two equations to eliminate the x term.
  • Not simplifying the equation and solving for x.
  • Not solving for y by isolating it on one side of the equation.
  • Not finding the values of x and y by substituting the values of x and y into one of the original equations.

Real-World Applications

Systems of equations with fractions have many real-world applications, including:

  • Physics: Systems of equations with fractions are used to describe the motion of objects in physics.
  • Engineering: Systems of equations with fractions are used to design and optimize systems in engineering.
  • Economics: Systems of equations with fractions are used to model economic systems and make predictions about economic trends.

Conclusion

In conclusion, solving systems of equations with fractions requires careful attention to detail and a solid understanding of algebraic methods. By following the steps outlined in this article, you can solve systems of equations with fractions and apply the techniques to real-world problems.

Introduction

Solving systems of equations with fractions can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will discuss some frequently asked questions about solving systems of equations with fractions.

Q: What is the first step in solving a system of equations with fractions?

A: The first step in solving a system of equations with fractions is to eliminate the fractions by multiplying both sides of each equation by the least common multiple (LCM) of the denominators.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you need to list the multiples of each denominator and find the smallest multiple that is common to both lists.

Q: What if the denominators are not the same?

A: If the denominators are not the same, you need to find the LCM of the two denominators. This can be done by listing the multiples of each denominator and finding the smallest multiple that is common to both lists.

Q: How do I add the two equations to eliminate the x term?

A: To add the two equations, you need to add the coefficients of the x terms and the constants on the right-hand side of the equations.

Q: What if the coefficients of the x terms are not the same?

A: If the coefficients of the x terms are not the same, you need to multiply the first equation by the coefficient of the x term in the second equation and the second equation by the coefficient of the x term in the first equation.

Q: How do I solve for x?

A: To solve for x, you need to isolate the x term on one side of the equation by adding or subtracting the same value from both sides of the equation.

Q: What if the equation is not linear?

A: If the equation is not linear, you need to use a different method to solve for x, such as substitution or elimination.

Q: How do I solve for y?

A: To solve for y, you need to isolate the y term on one side of the equation by adding or subtracting the same value from both sides of the equation.

Q: What if the equation is not linear?

A: If the equation is not linear, you need to use a different method to solve for y, such as substitution or elimination.

Q: Can I use a calculator to solve systems of equations with fractions?

A: Yes, you can use a calculator to solve systems of equations with fractions. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct formula.

Q: What are some common mistakes to avoid when solving systems of equations with fractions?

A: Some common mistakes to avoid when solving systems of equations with fractions include:

  • Not multiplying both sides of each equation by the least common multiple of the denominators.
  • Not using algebraic methods to solve the system of equations.
  • Not adding the two equations to eliminate the x term.
  • Not simplifying the equation and solving for x.
  • Not solving for y by isolating it on one side of the equation.

Q: What are some real-world applications of solving systems of equations with fractions?

A: Some real-world applications of solving systems of equations with fractions include:

  • Physics: Systems of equations with fractions are used to describe the motion of objects in physics.
  • Engineering: Systems of equations with fractions are used to design and optimize systems in engineering.
  • Economics: Systems of equations with fractions are used to model economic systems and make predictions about economic trends.

Conclusion

In conclusion, solving systems of equations with fractions requires careful attention to detail and a solid understanding of algebraic methods. By following the steps outlined in this article, you can solve systems of equations with fractions and apply the techniques to real-world problems.