Solve The System Of Equations And Choose The Correct Ordered Pair.$\[ \begin{array}{l} 3x + 2y = 12 \\ 6x + 3y = 21 \end{array} \\]A. \[$(4, 3)\$\] B. \[$(2, 3)\$\] C. \[$(3, 2)\$\] D. \[$(4, 0)\$\]

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Introduction

Solving systems of equations is a fundamental concept in mathematics, particularly in algebra. 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 provide a step-by-step solution to find the correct ordered pair.

The System of Equations

The given system of equations is:

3x+2y=126x+3y=21\begin{array}{l} 3x + 2y = 12 \\ 6x + 3y = 21 \end{array}

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. We can solve one equation for one variable and then substitute that expression into the other equation.

Step 1: Solve the first equation for x

We can solve the first equation for x by isolating x on one side of the equation.

3x+2y=123x + 2y = 12

Subtract 2y from both sides:

3x=12−2y3x = 12 - 2y

Divide both sides by 3:

x=12−2y3x = \frac{12 - 2y}{3}

Step 2: Substitute the expression for x into the second equation

Now that we have an expression for x, we can substitute it into the second equation.

6x+3y=216x + 3y = 21

Substitute the expression for x:

6(12−2y3)+3y=216\left(\frac{12 - 2y}{3}\right) + 3y = 21

Simplify the expression:

2(12−2y)+3y=212(12 - 2y) + 3y = 21

Expand and simplify:

24−4y+3y=2124 - 4y + 3y = 21

Combine like terms:

−y=−3-y = -3

Divide both sides by -1:

y=3y = 3

Step 3: Find the value of x

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

x=12−2y3x = \frac{12 - 2y}{3}

Substitute y = 3:

x=12−2(3)3x = \frac{12 - 2(3)}{3}

Simplify the expression:

x=12−63x = \frac{12 - 6}{3}

Combine like terms:

x=63x = \frac{6}{3}

Divide both sides by 3:

x=2x = 2

Step 4: Write the solution as an ordered pair

The solution to the system of equations is x = 2 and y = 3. We can write this as an ordered pair:

(2,3)(2, 3)

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. We can multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same.

Step 1: Multiply the first equation by 3 and the second equation by 2

Multiply the first equation by 3:

9x+6y=369x + 6y = 36

Multiply the second equation by 2:

12x+6y=4212x + 6y = 42

Step 2: Subtract the first equation from the second equation

Subtract the first equation from the second equation:

(12x+6y)−(9x+6y)=42−36(12x + 6y) - (9x + 6y) = 42 - 36

Simplify the expression:

3x=63x = 6

Divide both sides by 3:

x=2x = 2

Step 3: Find the value of y

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

3x+2y=123x + 2y = 12

Substitute x = 2:

3(2)+2y=123(2) + 2y = 12

Simplify the expression:

6+2y=126 + 2y = 12

Subtract 6 from both sides:

2y=62y = 6

Divide both sides by 2:

y=3y = 3

Step 4: Write the solution as an ordered pair

The solution to the system of equations is x = 2 and y = 3. We can write this as an ordered pair:

(2,3)(2, 3)

Conclusion

In this article, we solved a system of two linear equations with two variables using the substitution method and the elimination method. We found that the solution to the system of equations is x = 2 and y = 3, which can be written as the ordered pair (2, 3).

Answer

The correct ordered pair is:

(2,3)(2, 3)

Introduction

Solving systems of equations is a fundamental concept in mathematics, particularly in algebra. In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables. In this article, we will answer some frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that involve two or more variables. Each equation in the system is a statement that two expressions are equal.

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

There are two main methods to solve a system of equations: the substitution method and 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 multiplying the two equations by necessary multiples such that the coefficients of one variable are the same, and then subtracting one equation from the other.

Q: What is the difference between the substitution method and 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 multiplying the two equations by necessary multiples such that the coefficients of one variable are the same, and then subtracting one equation from the other.

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

A system of equations has a solution if and only if the two equations are consistent. This means that the two equations must have at least one solution in common.

Q: What is the solution to a system of equations?

The solution to a system of equations is the set of values that satisfy both equations. This can be written as an ordered pair (x, y).

Q: How do I write the solution to a system of equations as an ordered pair?

To write the solution to a system of equations as an ordered pair, you need to find the values of x and y that satisfy both equations. You can do this by solving one equation for one variable and then substituting that expression into the other equation.

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

If you have a system of equations with no solution, it means that the two equations are inconsistent. This can happen if the two equations are parallel lines that never intersect.

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

If you have a system of equations with infinitely many solutions, it means that the two equations are identical. This can happen if the two equations are the same line.

Q: How do I graph a system of equations?

To graph a system of equations, you need to graph each equation separately and then find the point of intersection. This can be done using a graphing calculator or by plotting the equations on a coordinate plane.

Q: What is the importance of solving systems of equations?

Solving systems of equations is an important concept in mathematics because it allows us to model real-world problems and find solutions to them. It is used in a wide range of fields, including physics, engineering, economics, and computer science.

Conclusion

In this article, we answered some frequently asked questions about solving systems of equations. We hope that this article has provided you with a better understanding of how to solve systems of equations and has helped you to become more confident in your ability to solve them.

Frequently Asked Questions

  • Q: What is a system of equations?
  • A: A system of equations is a set of two or more equations that involve two or more variables.
  • Q: How do I know which method to use to solve a system of equations?
  • A: There are two main methods to solve a system of equations: the substitution method and the elimination method.
  • 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 multiplying the two equations by necessary multiples such that the coefficients of one variable are the same, and then subtracting one equation from the other.
  • Q: How do I know if a system of equations has a solution?
  • A: A system of equations has a solution if and only if the two equations are consistent.
  • Q: What is the solution to a system of equations?
  • A: The solution to a system of equations is the set of values that satisfy both equations.
  • Q: How do I write the solution to a system of equations as an ordered pair?
  • A: To write the solution to a system of equations as an ordered pair, you need to find the values of x and y that satisfy both equations.
  • Q: What if I have a system of equations with no solution?
  • A: If you have a system of equations with no solution, it means that the two equations are inconsistent.
  • Q: What if I have a system of equations with infinitely many solutions?
  • A: If you have a system of equations with infinitely many solutions, it means that the two equations are identical.
  • Q: How do I graph a system of equations?
  • A: To graph a system of equations, you need to graph each equation separately and then find the point of intersection.
  • Q: What is the importance of solving systems of equations?
  • A: Solving systems of equations is an important concept in mathematics because it allows us to model real-world problems and find solutions to them.