Solve The System Of Equations:1. $-5x + 5y = -25$2. 3 X + 2 Y = 10 3x + 2y = 10 3 X + 2 Y = 10

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Introduction

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In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of two linear equations using the method of substitution and elimination. We will use the given equations:

1. $-5x + 5y = -25$
2. $3x + 2y = 10$

Understanding the System of Equations

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A system of linear equations is a set of equations that can be written in the form:

$ax + by = c$

where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. In this case, we have two equations:

1. $-5x + 5y = -25$
2. $3x + 2y = 10$

Method of Substitution

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The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. Let's solve the first equation for $y$:

$-5x + 5y = -25$

$5y = 5x - 25$

$y = x - 5$

Now, substitute this expression for $y$ into the second equation:

$3x + 2y = 10$

$3x + 2(x - 5) = 10$

$3x + 2x - 10 = 10$

$5x - 10 = 10$

$5x = 20$

$x = 4$

Finding the Value of y

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Now that we have found the value of $x$, we can substitute it into one of the original equations to find the value of $y$. Let's use the first equation:

$-5x + 5y = -25$

$-5(4) + 5y = -25$

$-20 + 5y = -25$

$5y = -5$

$y = -1$

Conclusion

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In this article, we have solved a system of two linear equations using the method of substitution. We first solved one equation for one variable and then substituted that expression into the other equation. We then found the value of the other variable by substituting the value of the first variable into one of the original equations. The final answer is:

$x = 4$

$y = -1$

Example Problems

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Problem 1

Solve the system of equations:

1. $2x + 3y = 7$
2. $x - 2y = -3$

Solution

To solve this system of equations, we can use the method of substitution. Let's solve the first equation for $y$:

$2x + 3y = 7$

$3y = 7 - 2x$

$y = \frac{7 - 2x}{3}$

Now, substitute this expression for $y$ into the second equation:

$x - 2y = -3$

$x - 2\left(\frac{7 - 2x}{3}\right) = -3$

$x - \frac{14 - 4x}{3} = -3$

$3x - 14 + 4x = -9$

$7x - 14 = -9$

$7x = 5$

$x = \frac{5}{7}$

Now, substitute this value of $x$ into the expression for $y$:

$y = \frac{7 - 2\left(\frac{5}{7}\right)}{3}$

$y = \frac{7 - \frac{10}{7}}{3}$

$y = \frac{\frac{49}{7} - \frac{10}{7}}{3}$

$y = \frac{\frac{39}{7}}{3}$

$y = \frac{39}{21}$

$y = \frac{13}{7}$

The final answer is:

$x = \frac{5}{7}$

$y = \frac{13}{7}$

Problem 2

Solve the system of equations:

1. $x + 2y = 6$
2. $2x - 3y = -2$

Solution

To solve this system of equations, we can use the method of elimination. Let's multiply the first equation by 2 and the second equation by 1:

$2(x + 2y) = 2(6)$

$2x + 4y = 12$

$2x - 3y = -2$

Now, add the two equations:

$(2x + 4y) + (2x - 3y) = 12 + (-2)$

$4x + y = 10$

Now, solve for $y$:

$y = 10 - 4x$

Now, substitute this expression for $y$ into one of the original equations. Let's use the first equation:

$x + 2y = 6$

$x + 2(10 - 4x) = 6$

$x + 20 - 8x = 6$

$-7x + 20 = 6$

$-7x = -14$

$x = 2$

Now, substitute this value of $x$ into the expression for $y$:

$y = 10 - 4(2)$

$y = 10 - 8$

$y = 2$

The final answer is:

$x = 2$

$y = 2$

Tips and Tricks

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Using the Method of Elimination

The method of elimination involves multiplying one or both equations by a constant to eliminate one of the variables. This can be done by multiplying the first equation by a constant that will make the coefficients of one of the variables the same in both equations.

Using the Method of Substitution

The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. This can be done by solving one equation for one variable and then substituting that expression into the other equation.

Checking the Solution

Once you have found the values of the variables, you should check the solution by substituting the values back into the original equations. If the solution is correct, the equations should be satisfied.

Conclusion

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In this article, we have solved a system of two linear equations using the method of substitution and elimination. We have also provided example problems and tips and tricks for solving systems of linear equations. The final answer is:

$x = 4$

$y = -1$

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Q: What is a system of linear equations?

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A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: What are the methods of solving systems of linear equations?

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A: There are two main methods of solving systems of linear equations: the method of substitution and the method of elimination.

Q: What is the method of substitution?

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A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the method of elimination?

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A: The method of elimination involves multiplying one or both equations by a constant to eliminate one of the variables.

Q: How do I choose which method to use?

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A: You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of one of the variables are the same in both equations, you can use the method of elimination. If the coefficients of one of the variables are different in both equations, you can use the method of substitution.

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

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

Q: How do I check my solution?

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A: To check your solution, you should substitute the values of the variables back into the original equations and make sure that the equations are satisfied.

Q: What if I have a system of linear equations with no solution?

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A: If you have a system of linear equations with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.

Q: What if I have a system of linear equations with infinitely many solutions?

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A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that can satisfy both equations.

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

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A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations.

Q: Can I use a computer program to solve systems of linear equations?

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A: Yes, you can use a computer program to solve systems of linear equations. Many computer programs, such as MATLAB and Python, have built-in functions for solving systems of linear equations.

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

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

• Not checking the solution
• Not using the correct method
• Not following the order of operations
• Not simplifying the equations

Q: How can I practice solving systems of linear equations?

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A: You can practice solving systems of linear equations by working through example problems and exercises. You can also use online resources, such as Khan Academy and Mathway, to practice solving systems of linear equations.

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

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

• Physics: Solving systems of linear equations is used to describe the motion of objects in physics.
• Engineering: Solving systems of linear equations is used to design and optimize systems in engineering.
• Economics: Solving systems of linear equations is used to model economic systems and make predictions about economic trends.
• Computer Science: Solving systems of linear equations is used in computer science to solve problems in machine learning and data analysis.

Q: Can I use solving systems of linear equations to solve other types of problems?

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A: Yes, you can use solving systems of linear equations to solve other types of problems, such as:

• Quadratic equations
• Polynomial equations
• Rational equations
• Trigonometric equations

Q: What are some advanced topics in solving systems of linear equations?

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A: Some advanced topics in solving systems of linear equations include:

• Matrix operations
• Determinants
• Eigenvalues and eigenvectors
• Linear transformations

Q: Can I use solving systems of linear equations to solve problems in other fields?

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A: Yes, you can use solving systems of linear equations to solve problems in other fields, such as:

• Biology: Solving systems of linear equations is used to model population growth and other biological systems.
• Chemistry: Solving systems of linear equations is used to model chemical reactions and other chemical systems.
• Medicine: Solving systems of linear equations is used to model medical systems and make predictions about patient outcomes.

Q: What are some resources for learning more about solving systems of linear equations?

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A: Some resources for learning more about solving systems of linear equations include:

• Khan Academy
• Mathway
• Wolfram Alpha
• MIT OpenCourseWare
• Coursera
• edX

Q: Can I use solving systems of linear equations to solve problems in other languages?

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A: Yes, you can use solving systems of linear equations to solve problems in other languages, such as:

• Spanish
• French
• German
• Chinese
• Japanese
• Arabic

Q: What are some common mistakes to avoid when translating mathematical problems?

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A: Some common mistakes to avoid when translating mathematical problems include:

• Not translating the problem correctly
• Not using the correct terminology
• Not following the correct order of operations
• Not simplifying the equations

Q: How can I practice translating mathematical problems?

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A: You can practice translating mathematical problems by working through example problems and exercises. You can also use online resources, such as Khan Academy and Mathway, to practice translating mathematical problems.

Q: What are some real-world applications of translating mathematical problems?

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A: Some real-world applications of translating mathematical problems include:

• International business: Translating mathematical problems is used to communicate mathematical ideas and solutions to people who speak different languages.
• Science: Translating mathematical problems is used to communicate scientific ideas and solutions to people who speak different languages.
• Engineering: Translating mathematical problems is used to communicate engineering ideas and solutions to people who speak different languages.
• Education: Translating mathematical problems is used to communicate mathematical ideas and solutions to students who speak different languages.

Q: Can I use translating mathematical problems to solve other types of problems?

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A: Yes, you can use translating mathematical problems to solve other types of problems, such as:

• Language translation
• Cultural translation
• Technical translation
• Business translation

Q: What are some advanced topics in translating mathematical problems?

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A: Some advanced topics in translating mathematical problems include:

• Machine translation
• Neural machine translation
• Statistical machine translation
• Post-editing

Q: Can I use translating mathematical problems to solve problems in other fields?

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A: Yes, you can use translating mathematical problems to solve problems in other fields, such as:

• Biology: Translating mathematical problems is used to model population growth and other biological systems.
• Chemistry: Translating mathematical problems is used to model chemical reactions and other chemical systems.
• Medicine: Translating mathematical problems is used to model medical systems and make predictions about patient outcomes.

Q: What are some resources for learning more about translating mathematical problems?

---

A: Some resources for learning more about translating mathematical problems include:

• Khan Academy
• Mathway
• Wolfram Alpha
• MIT OpenCourseWare
• Coursera
• edX

Q: Can I use translating mathematical problems to solve problems in other languages?

---

A: Yes, you can use translating mathematical problems to solve problems in other languages, such as:

• Spanish
• French
• German
• Chinese
• Japanese
• Arabic

Q: What are some common mistakes to avoid when using translating mathematical problems?

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A: Some common mistakes to avoid when using translating mathematical problems include:

• Not translating the problem correctly
• Not using the correct terminology
• Not following the correct order of operations
• Not simplifying the equations

Q: How can I practice using translating mathematical problems?

---

A: You can practice using translating mathematical problems by working through example problems and exercises. You can also use online resources, such as Khan Academy and Mathway, to practice using translating mathematical problems.

Q: What are some real-world applications of using translating mathematical problems?

---

A: Some real-world applications of using translating mathematical problems include:

• International business: Using translating mathematical problems is used to communicate mathematical ideas and solutions to people who speak different languages.
• Science: Using translating mathematical problems is used to communicate scientific ideas and solutions to people who speak different languages.
• Engineering: Using translating mathematical problems is used to communicate engineering ideas and solutions to people who speak different languages.
• Education: Using translating mathematical problems is used to communicate mathematical ideas and solutions to students who speak different languages.

Q: Can I use using translating mathematical problems to solve other types of problems?

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A: Yes, you can use using translating mathematical problems to solve other types of problems, such as:

• Language translation
• Cultural translation
• Technical translation
• Business translation

**Q: What are some advanced topics in using translating mathematical problems?