Solve The Following Pair Of Simultaneous Equations:$\[ \begin{array}{l} 5x - 7y + 3 = 0 \\ 3x + 2y + 8 = 0 \end{array} \\]

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Introduction

Simultaneous equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to solve a pair of simultaneous equations using the method of substitution and elimination. We will use a specific example to illustrate the process and provide a step-by-step guide on how to solve the equations.

What are Simultaneous Equations?

Simultaneous equations are a set of two or more equations that involve the same variables. In this case, we have two equations with two variables, x and y. The equations are:

{ \begin{array}{l} 5x - 7y + 3 = 0 \\ 3x + 2y + 8 = 0 \end{array} \}

The Method of Substitution

One way to solve simultaneous equations is by using the method of substitution. This involves solving one of the equations for one of the variables and then substituting that expression into the other equation.

Step 1: Solve One of the Equations for One of the Variables

Let's solve the first equation for x:

5x - 7y + 3 = 0

Subtract 3 from both sides:

5x - 7y = -3

Add 7y to both sides:

5x = 7y - 3

Divide both sides by 5:

x = (7y - 3) / 5

Step 2: Substitute the Expression into the Other Equation

Now that we have an expression for x, we can substitute it into the second equation:

3x + 2y + 8 = 0

Substitute x = (7y - 3) / 5 into the equation:

3((7y - 3) / 5) + 2y + 8 = 0

Step 3: Simplify the Equation

Simplify the equation by multiplying both sides by 5:

3(7y - 3) + 10y + 40 = 0

Expand the equation:

21y - 9 + 10y + 40 = 0

Combine like terms:

31y + 31 = 0

Subtract 31 from both sides:

31y = -31

Divide both sides by 31:

y = -1

Step 4: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y into one of the original equations. Let's use the first equation:

5x - 7y + 3 = 0

Substitute y = -1 into the equation:

5x - 7(-1) + 3 = 0

Simplify the equation:

5x + 7 + 3 = 0

Combine like terms:

5x + 10 = 0

Subtract 10 from both sides:

5x = -10

Divide both sides by 5:

x = -2

The Method of Elimination

Another way to solve simultaneous equations is by using the method of elimination. This involves adding or subtracting the equations to eliminate one of the variables.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples. Let's multiply the first equation by 2 and the second equation by 7:

10x - 14y + 6 = 0

21x + 14y + 56 = 0

Step 2: Add or Subtract the Equations

Now that we have the equations multiplied by necessary multiples, we can add or subtract them to eliminate one of the variables. Let's add the equations:

(10x - 14y + 6) + (21x + 14y + 56) = 0 + 0

Combine like terms:

31x + 62 = 0

Subtract 62 from both sides:

31x = -62

Divide both sides by 31:

x = -2

Step 3: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. Let's use the first equation:

5x - 7y + 3 = 0

Substitute x = -2 into the equation:

5(-2) - 7y + 3 = 0

Simplify the equation:

-10 - 7y + 3 = 0

Combine like terms:

-7 - 7y = 0

Add 7 to both sides:

-7y = 7

Divide both sides by -7:

y = -1

Conclusion

Solving simultaneous equations is a crucial skill for students and professionals alike. In this article, we explored two methods for solving simultaneous equations: the method of substitution and the method of elimination. We used a specific example to illustrate the process and provide a step-by-step guide on how to solve the equations. By following these steps, you can solve simultaneous equations and apply the concepts to real-world problems.

Final Answer

The final answer is:

x = -2

y = -1

Introduction

Simultaneous equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we explored two methods for solving simultaneous equations: the method of substitution and the method of elimination. In this article, we will answer some frequently asked questions about solving simultaneous equations.

Q: What are simultaneous equations?

A: Simultaneous equations are a set of two or more equations that involve the same variables. In this case, we have two equations with two variables, x and y.

Q: How do I know which method to use?

A: The choice of method depends on the equations and the variables involved. If the coefficients of one variable are the same in both equations, the method of elimination is usually the best choice. If the coefficients of one variable are different in both equations, the method of substitution is usually the best choice.

Q: What if I have three or more equations?

A: If you have three or more equations, you can use the method of substitution or elimination to solve for two variables, and then use the third equation to solve for the third variable.

Q: Can I use a calculator to solve simultaneous equations?

A: Yes, you can use a calculator to solve simultaneous equations. However, it's always a good idea to understand the underlying math and to check your answers to make sure they are correct.

Q: What if I get stuck or make a mistake?

A: Don't worry! Making mistakes is a normal part of the learning process. If you get stuck or make a mistake, try to identify where you went wrong and try again. You can also ask a teacher or tutor for help.

Q: Can I use simultaneous equations to solve real-world problems?

A: Yes, simultaneous equations can be used to solve a wide range of real-world problems, including:

  • Finance: Simultaneous equations can be used to model financial systems and make predictions about future stock prices or interest rates.
  • Engineering: Simultaneous equations can be used to design and optimize systems, such as bridges or buildings.
  • Science: Simultaneous equations can be used to model and analyze complex systems, such as the behavior of particles in a gas or the spread of diseases.

Q: What are some common mistakes to avoid when solving simultaneous equations?

A: Some common mistakes to avoid when solving simultaneous equations include:

  • Not checking your work: Make sure to check your answers to make sure they are correct.
  • Not using the correct method: Choose the method that is best suited to the equations and variables involved.
  • Not simplifying the equations: Simplify the equations as much as possible to make them easier to solve.

Conclusion

Solving simultaneous equations is a crucial skill for students and professionals alike. By understanding the underlying math and using the right methods, you can solve simultaneous equations and apply the concepts to real-world problems. Remember to check your work, choose the right method, and simplify the equations to avoid common mistakes.

Additional Resources

If you're looking for additional resources to help you learn about solving simultaneous equations, here are a few suggestions:

  • Textbooks: There are many excellent textbooks on mathematics that cover simultaneous equations, including "Algebra" by Michael Artin and "Linear Algebra and Its Applications" by Gilbert Strang.
  • Online resources: There are many online resources available to help you learn about solving simultaneous equations, including Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
  • Practice problems: Practice problems are a great way to reinforce your understanding of solving simultaneous equations. You can find practice problems online or in textbooks.

Final Answer

The final answer is:

  • x = -2
  • y = -1

Note: The final answer is the solution to the pair of simultaneous equations.