Solve The Following System Of Equations:$\[ \begin{array}{l} 4x + 3y = 2 \\ 7x - 2y = -11 \end{array} \\]\[$ X = \$\]\[$ Y = \$\]

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.

The System of Equations

The system of equations we will be solving is:

4x+3y=27x−2y=−11{ \begin{array}{l} 4x + 3y = 2 \\ 7x - 2y = -11 \end{array} }

Method 1: Substitution Method

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

Step 1: Solve the First Equation for x

We can solve the first equation for x by subtracting 3y from both sides:

4x=2−3y{ 4x = 2 - 3y }

Then, we can divide both sides by 4:

x=2−3y4{ x = \frac{2 - 3y}{4} }

Step 2: Substitute the Expression for x into the Second Equation

Now, we can substitute the expression for x into the second equation:

7(2−3y4)−2y=−11{ 7\left(\frac{2 - 3y}{4}\right) - 2y = -11 }

Step 3: Simplify the Equation

We can simplify the equation by multiplying both sides by 4:

7(2−3y)−8y=−44{ 7(2 - 3y) - 8y = -44 }

Expanding the left-hand side, we get:

14−21y−8y=−44{ 14 - 21y - 8y = -44 }

Combining like terms, we get:

−29y=−58{ -29y = -58 }

Step 4: Solve for y

We can solve for y by dividing both sides by -29:

y=−58−29{ y = \frac{-58}{-29} }

y=2{ y = 2 }

Step 5: Find the Value of x

Now that we have the value of y, we can substitute it into one of the original equations to find the value of x. We will use the first equation:

4x+3(2)=2{ 4x + 3(2) = 2 }

4x+6=2{ 4x + 6 = 2 }

Subtracting 6 from both sides, we get:

4x=−4{ 4x = -4 }

Dividing both sides by 4, we get:

x=−1{ x = -1 }

Method 2: Elimination Method

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

Step 1: Multiply the Equations by Necessary Multiples

We can multiply the first equation by 2 and the second equation by 3 to make the coefficients of y opposites:

8x+6y=4{ 8x + 6y = 4 }

21x−6y=−33{ 21x - 6y = -33 }

Step 2: Add the Equations

We can add the equations to eliminate the y-variable:

(8x+6y)+(21x−6y)=4+(−33){ (8x + 6y) + (21x - 6y) = 4 + (-33) }

29x=−29{ 29x = -29 }

Step 3: Solve for x

We can solve for x by dividing both sides by 29:

x=−2929{ x = \frac{-29}{29} }

x=−1{ x = -1 }

Step 4: Find the Value of y

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y. We will use the first equation:

4(−1)+3y=2{ 4(-1) + 3y = 2 }

−4+3y=2{ -4 + 3y = 2 }

Adding 4 to both sides, we get:

3y=6{ 3y = 6 }

Dividing both sides by 3, we get:

y=2{ y = 2 }

Conclusion

In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. Both methods have led to the same solution: x = -1 and y = 2.

Final Answer

The final answer is:

x=−1{ x = -1 }

y=2{ y = 2 }

Discussion

This system of equations can be solved using other methods as well, such as the graphing method or the matrix method. However, the substitution method and the elimination method are two of the most common methods used to solve systems of linear equations.

Real-World Applications

Systems of linear equations have many real-world applications, such as:

  • Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, the flow of fluids, and the stress on structures.
  • Economics: Systems of linear equations are used to model economic systems, such as the supply and demand of goods and services.
  • Computer Science: Systems of linear equations are used in computer science to solve problems, such as the shortest path problem and the minimum spanning tree problem.

Tips and Tricks

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

  • Use the substitution method when one of the equations is easy to solve for one variable.
  • Use the elimination method when the coefficients of one of the variables are opposites.
  • Check your work by plugging the values of x and y back into the original equations.
  • Use a graphing calculator or a computer program to visualize the system of equations and find the solution.

Practice Problems

Here are some practice problems to help you practice solving systems of linear equations:

  • Solve the system of equations:

2x+3y=5x−2y=−3{ \begin{array}{l} 2x + 3y = 5 \\ x - 2y = -3 \end{array} }

  • Solve the system of equations:

x+2y=43x−2y=2{ \begin{array}{l} x + 2y = 4 \\ 3x - 2y = 2 \end{array} }

  • Solve the system of equations:

x−3y=−22x+5y=11{ \begin{array}{l} x - 3y = -2 \\ 2x + 5y = 11 \end{array} }

Conclusion

Solving systems of linear equations is an important skill in mathematics and has many real-world applications. In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. We have also provided some tips and tricks to help you solve systems of linear equations and some practice problems to help you practice.

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Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables.

Q: How do I know which method to use to solve a system of linear equations?

A: You can use either the substitution method or the elimination method to solve a system of linear equations. The substitution method is often easier to use when one of the equations is easy to solve for one variable, while the elimination method is often easier to use when the coefficients of one of the variables are opposites.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I check my work when solving a system of linear equations?

A: You can check your work by plugging the values of x and y back into the original equations. If the values satisfy both equations, then you have found the correct solution.

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

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

  • Not checking your work: Make sure to plug the values of x and y back into the original equations to check that they satisfy both equations.
  • Not using the correct method: Make sure to use either the substitution method or the elimination method, depending on which one is easier to use.
  • Not simplifying the equations: Make sure to simplify the equations as much as possible before solving for the variables.

Q: How do I use a graphing calculator or a computer program to solve a system of linear equations?

A: You can use a graphing calculator or a computer program to visualize the system of equations and find the solution. To do this, you will need to enter the equations into the calculator or program and then use the built-in functions to find the solution.

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

A: Systems of linear equations have many real-world applications, including:

  • Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, the flow of fluids, and the stress on structures.
  • Economics: Systems of linear equations are used to model economic systems, such as the supply and demand of goods and services.
  • Computer Science: Systems of linear equations are used in computer science to solve problems, such as the shortest path problem and the minimum spanning tree problem.

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

A: You can practice solving systems of linear equations by working through practice problems, such as the ones provided in this article. You can also try solving systems of linear equations on your own, using either the substitution method or the elimination method.

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

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

  • Use the substitution method when one of the equations is easy to solve for one variable.
  • Use the elimination method when the coefficients of one of the variables are opposites.
  • Check your work by plugging the values of x and y back into the original equations.
  • Use a graphing calculator or a computer program to visualize the system of equations and find the solution.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if and only if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system of linear equations has no solution.

Q: What is the difference between a consistent system of linear equations and an inconsistent system of linear equations?

A: A consistent system of linear equations is a system of linear equations that has a solution, while an inconsistent system of linear equations is a system of linear equations that has no solution.

Q: How do I determine if a system of linear equations is consistent or inconsistent?

A: You can determine if a system of linear equations is consistent or inconsistent by checking if the two equations are parallel or not. If the two equations are parallel, then the system of linear equations is inconsistent. If the two equations are not parallel, then the system of linear equations is consistent.

Q: What are some common mistakes to avoid when determining if a system of linear equations is consistent or inconsistent?

A: Some common mistakes to avoid when determining if a system of linear equations is consistent or inconsistent include:

  • Not checking if the two equations are parallel: Make sure to check if the two equations are parallel before determining if the system of linear equations is consistent or inconsistent.
  • Not using the correct method: Make sure to use the correct method to determine if the system of linear equations is consistent or inconsistent.
  • Not simplifying the equations: Make sure to simplify the equations as much as possible before determining if the system of linear equations is consistent or inconsistent.