Solve The System Of Equations:${ \begin{array}{l} -\frac{2}{3} X + 3y = -34 \ x = -3y + 3 \end{array} }$

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Introduction

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Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It 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. We will use the given system of equations as an example and walk through the steps to find the solution.

The System of Equations

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The given system of equations is:

${ \begin{array}{l} -\frac{2}{3} x + 3y = -34 \\ x = -3y + 3 \end{array} \}$

Step 1: Write Down the Equations

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The first step is to write down the given equations. We have two equations:

1. $-\frac{2}{3} x + 3y = -34$
2. $x = -3y + 3$

Step 2: Solve One Equation for One Variable

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We will solve the second equation for $x$ in terms of $y$. This will give us:

$$x = -3y + 3$$

Step 3: Substitute the Expression into the Other Equation

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We will substitute the expression for $x$ into the first equation:

$$-\frac{2}{3} (-3y + 3) + 3y = -34$$

Step 4: Simplify the Equation

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We will simplify the equation by distributing the negative sign and combining like terms:

$$2y - 2 + 3y = -34$$

Step 5: Combine Like Terms

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We will combine the like terms:

$$5y - 2 = -34$$

Step 6: Add 2 to Both Sides

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We will add 2 to both sides of the equation:

$$5y = -32$$

Step 7: Divide Both Sides by 5

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We will divide both sides of the equation by 5:

$$y = -\frac{32}{5}$$

Step 8: Substitute the Value of y into One of the Original Equations

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We will substitute the value of $y$ into the second original equation:

$$x = -3(-\frac{32}{5}) + 3$$

Step 9: Simplify the Equation

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We will simplify the equation:

$$x = \frac{96}{5} + 3$$

Step 10: Combine Like Terms

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We will combine the like terms:

$$x = \frac{96 + 15}{5}$$

Step 11: Simplify the Fraction

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We will simplify the fraction:

$$x = \frac{111}{5}$$

Conclusion

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We have solved the system of linear equations using the substitution method. The solution is $x = \frac{111}{5}$ and $y = -\frac{32}{5}$.

Final Answer

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The final answer is $\boxed{x = \frac{111}{5}, y = -\frac{32}{5}}$.

Why is this Method Important?

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Solving systems of linear equations is an essential skill in mathematics, particularly in algebra and geometry. It has numerous applications in various fields, such as physics, engineering, economics, and computer science. The substitution method is one of the most common methods used to solve systems of linear equations. It is a powerful tool that allows us to find the solution to a system of equations by substituting one equation into another.

Real-World Applications

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Solving systems of linear equations has numerous real-world applications. For example:

• In physics, systems of linear equations are used to describe the motion of objects under the influence of forces.
• In engineering, systems of linear equations are used to design and analyze electrical circuits, mechanical systems, and structural systems.
• In economics, systems of linear equations are used to model the behavior of economic systems, such as supply and demand curves.
• In computer science, systems of linear equations are used in machine learning and data analysis.

Tips and Tricks

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

• Use the substitution method to solve systems of linear equations.
• Simplify the equations by combining like terms.
• Use fractions to simplify the equations.
• Check your solution by plugging it back into the original equations.

Common Mistakes

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

• Not simplifying the equations by combining like terms.
• Not using fractions to simplify the equations.
• Not checking the solution by plugging it back into the original equations.
• Not using the substitution method to solve the system of equations.

Conclusion

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Solving systems of linear equations is an essential skill in mathematics, particularly in algebra and geometry. The substitution method is one of the most common methods used to solve systems of linear equations. It is a powerful tool that allows us to find the solution to a system of equations by substituting one equation into another. With practice and patience, you can master the art of solving systems of linear equations and apply it to real-world problems.

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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 different methods to solve a system of linear equations?

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A: There are several methods to solve a system of linear equations, including:

• Substitution method: This method involves solving one equation for one variable and substituting it into the other equation.
• Elimination method: This method involves adding or subtracting the equations to eliminate one of the variables.
• Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
• Matrix method: This method involves using matrices to solve the system of equations.

Q: What is the substitution method?

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A: The substitution method is a method of solving a system of linear equations by solving one equation for one variable and substituting it into the other equation.

Q: How do I use the substitution method to solve a system of linear equations?

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A: To use the substitution method, follow these steps:

1. Solve one equation for one variable.
2. Substitute the expression into the other equation.
3. Simplify the equation and solve for the other variable.
4. Check the solution by plugging it back into the original equations.

Q: What is the elimination method?

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A: The elimination method is a method of solving a system of linear equations by adding or subtracting the equations to eliminate one of the variables.

Q: How do I use the elimination method to solve a system of linear equations?

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A: To use the elimination method, follow these steps:

1. Multiply the equations by necessary multiples such that the coefficients of one of the variables are the same in both equations.
2. Add or subtract the equations to eliminate one of the variables.
3. Simplify the equation and solve for the other variable.
4. Check the solution by plugging it back into the original equations.

Q: What is the graphical method?

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A: The graphical method is a method of solving a system of linear equations by graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I use the graphical method to solve a system of linear equations?

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A: To use the graphical method, follow these steps:

1. Graph the equations on a coordinate plane.
2. Find the point of intersection.
3. Check the solution by plugging it back into the original equations.

Q: What is the matrix method?

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A: The matrix method is a method of solving a system of linear equations by using matrices to solve the system of equations.

Q: How do I use the matrix method to solve a system of linear equations?

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A: To use the matrix method, follow these steps:

1. Write the system of equations in matrix form.
2. Use the inverse matrix to solve the system of equations.
3. Check the solution by plugging it back into the original 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 simplifying the equations by combining like terms.
• Not using fractions to simplify the equations.
• Not checking the solution by plugging it back into the original equations.
• Not using the correct method to solve the system of equations.

Q: How do I check my solution to a system of linear equations?

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A: To check your solution to a system of linear equations, follow these steps:

1. Plug the solution back into the original equations.
2. Check if the solution satisfies both equations.
3. If the solution satisfies both equations, then it is the correct solution.

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:

• In physics, systems of linear equations are used to describe the motion of objects under the influence of forces.
• In engineering, systems of linear equations are used to design and analyze electrical circuits, mechanical systems, and structural systems.
• In economics, systems of linear equations are used to model the behavior of economic systems, such as supply and demand curves.
• In computer science, systems of linear equations are used in machine learning and data analysis.

Q: How do I choose the correct method to solve a system of linear equations?

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A: To choose the correct method to solve a system of linear equations, follow these steps:

1. Look at the equations and determine which method is most suitable.
2. Consider the complexity of the equations and the number of variables.
3. Choose the method that is most efficient and effective.

Q: What are some tips and tricks for solving systems of linear equations?

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A: Some tips and tricks for solving systems of linear equations include:

• Use the substitution method to solve systems of linear equations.
• Simplify the equations by combining like terms.
• Use fractions to simplify the equations.
• Check the solution by plugging it back into the original equations.

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

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

• Not simplifying the equations by combining like terms.
• Not using fractions to simplify the equations.
• Not checking the solution by plugging it back into the original equations.
• Not using the correct method to solve the system of equations.

Conclusion

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Solving systems of linear equations is an essential skill in mathematics, particularly in algebra and geometry. The substitution method, elimination method, graphical method, and matrix method are all effective methods for solving systems of linear equations. By following the steps outlined in this article, you can master the art of solving systems of linear equations and apply it to real-world problems.