Which Ordered Pair { (a, B)$}$ Is The Solution To The Given System Of Linear Equations?${ \begin{aligned} 3a + B &= 10 \ -4a - 2b &= 2 \end{aligned} }$A. (1, 7) B. (3, 1) C. (11, -23) D. (23, -11)

by ADMIN 201 views

Introduction

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 linear equations simultaneously. In this article, we will explore how to solve a system of linear equations using the method of substitution and elimination. We will also apply this method to a specific system of linear equations to find the ordered pair that satisfies the given equations.

What is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve the same variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. The goal is to find the values of x and y that satisfy all the equations in the system.

Methods for Solving a System of Linear Equations

There are several methods for solving a system of linear equations, including:

  • Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination Method: This method involves adding or subtracting the equations to eliminate one variable and then solving for the other variable.
  • Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Solving the Given System of Linear Equations

The given system of linear equations is:

{ \begin{aligned} 3a + b &= 10 \\ -4a - 2b &= 2 \end{aligned} \}

To solve this system, we will use the elimination method. We will multiply the first equation by 2 and the second equation by 1 to make the coefficients of b in both equations equal.

{ \begin{aligned} 6a + 2b &= 20 \\ -4a - 2b &= 2 \end{aligned} \}

Now, we will add both equations to eliminate the variable b.

{ \begin{aligned} (6a + 2b) + (-4a - 2b) &= 20 + 2 \\ 2a &= 22 \end{aligned} \}

Next, we will solve for a by dividing both sides of the equation by 2.

{ \begin{aligned} 2a &= 22 \\ a &= \frac{22}{2} \\ a &= 11 \end{aligned} \}

Now that we have found the value of a, we can substitute it into one of the original equations to find the value of b. We will use the first equation.

{ \begin{aligned} 3a + b &= 10 \\ 3(11) + b &= 10 \\ 33 + b &= 10 \\ b &= -23 \end{aligned} \}

Therefore, the ordered pair that satisfies the given system of linear equations is (11, -23).

Conclusion

Solving a system of linear equations is a crucial concept in mathematics, and it has numerous applications in various fields, including science, engineering, and economics. In this article, we have explored how to solve a system of linear equations using the method of substitution and elimination. We have also applied this method to a specific system of linear equations to find the ordered pair that satisfies the given equations. By following the steps outlined in this article, you can solve any system of linear equations and find the values of variables that satisfy the given equations.

Final Answer

The final answer is (11, -23).

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it can be a bit challenging for some students. In this article, we will address some of the most frequently asked questions about solving systems of linear equations. We will cover topics such as the methods of substitution and elimination, how to choose the correct method, and how to solve systems with multiple variables.

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

A: 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 variable and then solving for the other variable.

Q: How do I choose the correct method?

A: To choose the correct method, you need to look at the coefficients of the variables in the equations. If the coefficients of one variable are the same, you can use the elimination method. If the coefficients of one variable are different, you can use the substitution method.

Q: What if I have a system with multiple variables?

A: If you have a system with multiple variables, you can use the same methods as before. However, you may need to use a combination of both methods to solve the system.

Q: How do I know if I have a consistent or inconsistent system?

A: To determine if you have a consistent or inconsistent system, you need to check if the equations are parallel or not. If the equations are parallel, the system is inconsistent. If the equations are not parallel, the system is consistent.

Q: What is the graphical method?

A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful for systems with two variables.

Q: How do I graph the equations?

A: To graph the equations, you need to plot the x and y intercepts of each equation. Then, you need to draw a line through the intercepts to represent the equation.

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

A: If you have a system with no solution, it means that the equations are inconsistent. In this case, you can say that the system has no solution.

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

A: If you have a system with infinitely many solutions, it means that the equations are dependent. In this case, you can say that the system has infinitely many solutions.

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

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.

Conclusion

Solving systems of linear equations can be a bit challenging, but with practice and patience, you can master it. In this article, we have addressed some of the most frequently asked questions about solving systems of linear equations. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Tips

  • Always read the problem carefully and understand what is being asked.
  • Choose the correct method for solving the system.
  • Check your work by plugging the solution back into the original equations.
  • Practice, practice, practice! The more you practice, the better you will become at solving systems of linear equations.