Solve The System Of Equations Below.$\[ \begin{array}{l} x+y=7 \\ 2x+3y=16 \end{array} \\]A. (5, 2)B. (2, 5)C. (3, 4)D. (4, 3)

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Introduction

In mathematics, a system of linear equations is a set of two or more equations in which the unknowns are related through linear relationships. Solving a system of linear equations involves finding the values of the unknowns that satisfy all the equations simultaneously. 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 given by:

x+y=72x+3y=16\begin{array}{l} x+y=7 \\ 2x+3y=16 \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 equation 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 y from both sides:

x=7−yx = 7 - y

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

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

2(7−y)+3y=162(7 - y) + 3y = 16

Step 3: Simplify the Equation

Expanding and simplifying the equation, we get:

14−2y+3y=1614 - 2y + 3y = 16

Step 4: Combine Like Terms

Combining like terms, we get:

14+y=1614 + y = 16

Step 5: Solve for y

Subtracting 14 from both sides, we get:

y=2y = 2

Step 6: Find the Value of x

Now that we have the value of y, we can substitute it back into the expression for x:

x=7−2x = 7 - 2

x=5x = 5

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 First Equation by 2

We can multiply the first equation by 2 to make the coefficients of x in both equations equal:

2x+2y=142x + 2y = 14

Step 2: Subtract the Second Equation from the First Equation

Now, we can subtract the second equation from the first equation:

(2x+2y)−(2x+3y)=14−16(2x + 2y) - (2x + 3y) = 14 - 16

Step 3: Simplify the Equation

Simplifying the equation, we get:

−y=−2-y = -2

Step 4: Solve for y

Dividing both sides by -1, we get:

y=2y = 2

Step 5: Find the Value of x

Now that we have the value of y, we can substitute it back into one of the original equations to find the value of x:

x+2=7x + 2 = 7

x=5x = 5

Conclusion

In this article, we have solved a system of two linear equations with two variables using the substitution method and the elimination method. We have found that the solution to the system is x = 5 and y = 2.

Answer

The correct answer is:

  • A. (5, 2)

Discussion

This system of equations can be solved using either the substitution method or the elimination method. The substitution method involves solving one equation 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.

Tips and Tricks

  • When solving a system of linear equations, it is often helpful to use the substitution method or the elimination method.
  • When using the substitution method, make sure to solve one equation for one variable before substituting that expression into the other equation.
  • When using the elimination method, make sure to add or subtract the equations in a way that eliminates one of the variables.

Related Topics

  • Solving systems of linear equations with three variables
  • Solving systems of linear equations with more than three variables
  • Solving systems of nonlinear equations

References

  • [1] "Solving Systems of Linear Equations" by Math Open Reference
  • [2] "Solving Systems of Linear Equations" by Khan Academy
    Solving Systems of Linear Equations: Q&A =============================================

Introduction

In our previous article, we discussed how to solve a system of two linear equations with two variables using the substitution method and the elimination method. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more equations in which the unknowns are related through linear relationships.

Q: How do I know which method to use to solve a system of linear 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, the elimination method is often the easiest to use. If the coefficients of one variable are different in both equations, the substitution method may be more convenient.

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

A: To solve a system of linear equations with three variables, you can use the same methods as before, but you will need to solve two equations for two variables before substituting that expression into the third equation.

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

A: Yes, you can use a graphing calculator to solve a system of linear equations. Simply graph the two equations on the same coordinate plane and find the point of intersection.

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

A: If you have a system of linear equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory, such as 2x + 3y = 5 and 2x + 3y = 10.

Q: Can I use a system of linear equations to model real-world problems?

A: Yes, systems of linear equations can be used to model a wide range of real-world problems, such as:

  • Finding the cost of producing a certain number of items
  • Determining the amount of money in a bank account
  • Calculating the area of a rectangle

Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?

A: To determine the number of solutions to a system of linear equations, you can use the following criteria:

  • If the equations are consistent and the number of equations is equal to the number of variables, there is a unique solution.
  • If the equations are inconsistent, there is no solution.
  • If the number of equations is greater than the number of variables, there are infinitely many solutions.

Q: Can I use a system of linear equations to solve a problem with more than two variables?

A: Yes, you can use a system of linear equations to solve a problem with more than two variables. However, you will need to use a more advanced method, such as the Gaussian elimination method or the matrix method.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the different methods for solving systems of linear equations, including the substitution method and the elimination method. We have also talked about how to use a graphing calculator to solve a system of linear equations and how to model real-world problems using systems of linear equations.

Tips and Tricks

  • When solving a system of linear equations, make sure to check your work by plugging the solution back into the original equations.
  • When using a graphing calculator to solve a system of linear equations, make sure to graph the equations on the same coordinate plane.
  • When modeling real-world problems using systems of linear equations, make sure to check your work by plugging the solution back into the original problem.

Related Topics

  • Solving systems of linear equations with three variables
  • Solving systems of linear equations with more than three variables
  • Solving systems of nonlinear equations

References

  • [1] "Solving Systems of Linear Equations" by Math Open Reference
  • [2] "Solving Systems of Linear Equations" by Khan Academy