Find The Values Of $x$ And $y$ In The Following System Of Equations:$ \begin{array}{l} \frac{x}{5}+\frac{2y}{3}=\frac{49}{15} \\ \frac{3x}{7}-\frac{y}{2}+\frac{5}{7}=0 \end{array} $

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Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables, x and y. We will use the given system of equations to demonstrate the steps involved in solving such a system.

The System of Equations

The given system of equations is:

x5+2y3=49153x7−y2+57=0\begin{array}{l} \frac{x}{5}+\frac{2y}{3}=\frac{49}{15} \\ \frac{3x}{7}-\frac{y}{2}+\frac{5}{7}=0 \end{array}

Step 1: Multiply Both Sides of Each Equation by the Least Common Multiple (LCM) of the Denominators

To eliminate the fractions, we need to multiply both sides of each equation by the LCM of the denominators. The LCM of 5, 3, and 15 is 15, and the LCM of 7, 2, and 7 is 14.

Multiplying the first equation by 15, we get:

15x5+30y3=49\frac{15x}{5}+\frac{30y}{3}=49

Simplifying the equation, we get:

3x+10y=493x+10y=49

Multiplying the second equation by 14, we get:

42x7−14y2+707=0\frac{42x}{7}-\frac{14y}{2}+\frac{70}{7}=0

Simplifying the equation, we get:

6x−7y+10=06x-7y+10=0

Step 2: Multiply the First Equation by 7 and the Second Equation by 10

To make the coefficients of y in both equations equal, we need to multiply the first equation by 7 and the second equation by 10.

Multiplying the first equation by 7, we get:

21x+70y=34321x+70y=343

Multiplying the second equation by 10, we get:

60x−70y+100=060x-70y+100=0

Step 3: Add Both Equations to Eliminate the Variable y

Now that the coefficients of y in both equations are equal, we can add both equations to eliminate the variable y.

Adding both equations, we get:

81x+100=34381x+100=343

Step 4: Solve for x

Now that we have eliminated the variable y, we can solve for x.

Subtracting 100 from both sides of the equation, we get:

81x=24381x=243

Dividing both sides of the equation by 81, we get:

x=24381x=\frac{243}{81}

Simplifying the equation, we get:

x=3x=3

Step 5: Substitute the Value of x into One of the Original Equations to Solve for y

Now that we have found the value of x, we can substitute it into one of the original equations to solve for y.

Substituting x=3 into the first original equation, we get:

35+2y3=4915\frac{3}{5}+\frac{2y}{3}=\frac{49}{15}

Multiplying both sides of the equation by 15, we get:

9+10y=499+10y=49

Subtracting 9 from both sides of the equation, we get:

10y=4010y=40

Dividing both sides of the equation by 10, we get:

y=4y=4

Conclusion

In this article, we have demonstrated the steps involved in solving a system of two linear equations with two variables, x and y. We have used the given system of equations to illustrate the process of solving such a system. By following the steps outlined in this article, you should be able to solve similar systems of equations.

Final Answer

The final answer is x=3 and y=4.

Additional Tips and Resources

  • To solve a system of equations, you can use the method of substitution or the method of elimination.
  • The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.
  • The method of elimination involves adding or subtracting the equations to eliminate one variable.
  • You can also use matrices to solve a system of equations.
  • For more information on solving systems of equations, you can refer to the following resources:
    • Khan Academy: Solving Systems of Equations
    • Mathway: Solving Systems of Equations
    • Wolfram Alpha: Solving Systems of Equations

Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will provide a Q&A guide to help you understand the process of solving a system of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve two or more variables. The goal is to find the values of the variables that satisfy all the equations simultaneously.

Q: What are the different methods for solving a system of equations?

A: There are two main methods for solving a system of equations: the method of substitution and the method of elimination. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.

Q: What is the least common multiple (LCM) and how is it used in solving a system of equations?

A: The LCM is the smallest multiple that is a common multiple of two or more numbers. In solving a system of equations, the LCM is used to eliminate fractions by multiplying both sides of each equation by the LCM.

Q: How do I know which method to use when solving a system of equations?

A: The choice of method depends on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can use the method of elimination. If the coefficients of one variable are different in both equations, you can use the method of substitution.

Q: What if I have a system of equations with three or more variables?

A: If you have a system of equations with three or more variables, you can use the method of substitution or the method of elimination to solve for two variables, and then substitute those values into one of the original equations to solve for the third variable.

Q: Can I use matrices to solve a system of equations?

A: Yes, you can use matrices to solve a system of equations. Matrices provide a concise and efficient way to represent the system of equations and solve for the variables.

Q: What are some common mistakes to avoid when solving a system of equations?

A: Some common mistakes to avoid when solving a system of equations include:

  • Not checking the validity of the solutions
  • Not using the correct method for the given system of equations
  • Not simplifying the equations before solving
  • Not checking for extraneous solutions

Q: Where can I find more resources for learning about solving systems of equations?

A: There are many resources available for learning about solving systems of equations, including:

  • Khan Academy: Solving Systems of Equations
  • Mathway: Solving Systems of Equations
  • Wolfram Alpha: Solving Systems of Equations
  • Online textbooks and tutorials
  • Math education websites and forums

Conclusion

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. By following the steps outlined in this Q&A guide, you should be able to solve systems of equations with confidence.

Final Tips

  • Practice solving systems of equations with different types of equations and variables.
  • Use matrices to solve systems of equations for a more efficient and concise solution.
  • Check the validity of the solutions and avoid extraneous solutions.
  • Use online resources and tutorials to supplement your learning and stay up-to-date with the latest techniques and methods.

Additional Resources

  • Khan Academy: Solving Systems of Equations
  • Mathway: Solving Systems of Equations
  • Wolfram Alpha: Solving Systems of Equations
  • Online textbooks and tutorials
  • Math education websites and forums