Checking Other Ordered Pairs

Oct 13, 2025 by talynt.org 149 views

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving a system of two linear equations with two variables. We will use the substitution method to find the solution to the system of equations.

The System of Equations

The system of equations we will be working with is:

{ \left\{ \begin{array}{l} 6x + 2y = 16 \\ y = -3x + 8 \end{array} \right. \}

Step 1: Write Down the Equations

The first equation is $6x + 2y = 16$ , and the second equation is $y = -3x + 8$ .

Step 2: Solve One of the Equations for One Variable

We will solve the second equation for $y$ :

$y = -3x + 8$

Step 3: Substitute the Expression into the Other Equation

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

$6x + 2(-3x + 8) = 16$

Step 4: Simplify the Equation

We will simplify the equation:

$6x - 6x + 16 = 16$

$16 = 16$

Step 5: Check the Solution

We have found that the equation is an identity, which means that it is true for all values of $x$ . However, we need to check if the solution satisfies both equations.

We will substitute $x = 0$ into the second equation:

$y = -3(0) + 8$

$y = 8$

We will substitute $x = 0$ and $y = 8$ into the first equation:

$6(0) + 2(8) = 16$

$16 = 16$

Conclusion

We have found that the ordered pair $(0, 8)$ is a solution to the system of equations.

Checking Other Ordered Pairs

We will check if the ordered pairs $(1, 2)$ , $(2, 4)$ , and $(3, 6)$ are solutions to the system of equations.

Ordered Pair (1, 2)

We will substitute $x = 1$ and $y = 2$ into the first equation:

$6(1) + 2(2) = 16$

$8 = 16$

The ordered pair $(1, 2)$ is not a solution to the system of equations.

Ordered Pair (2, 4)

We will substitute $x = 2$ and $y = 4$ into the first equation:

$6(2) + 2(4) = 16$

$20 = 16$

The ordered pair $(2, 4)$ is not a solution to the system of equations.

Ordered Pair (3, 6)

We will substitute $x = 3$ and $y = 6$ into the first equation:

$6(3) + 2(6) = 16$

$30 = 16$

The ordered pair $(3, 6)$ is not a solution to the system of equations.

Conclusion

We have found that the ordered pair $(0, 8)$ is a solution to the system of equations, and the ordered pairs $(1, 2)$ , $(2, 4)$ , and $(3, 6)$ are not solutions to the system of equations.

Final Answer

$The final answer is $\boxed{(0, 8)}$.<br/> **Solving Systems of Equations: A Q&A Guide**$

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will provide a Q&A guide to help you understand the concept of solving systems of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that are related to each other. Each equation contains one or more variables, and the goal is to find the values of the variables that satisfy all the equations.

Q: What are the different methods for solving systems of equations?

There are several methods for solving systems of 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 of the variables.
Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. For example, if we have the equations:

$2x + 3y = 7$

$y = 2x - 1$

We can solve the second equation for $y$ and then substitute that expression into the first equation.

Q: What is the elimination method?

The elimination method involves adding or subtracting the equations to eliminate one of the variables. For example, if we have the equations:

$2x + 3y = 7$

$4x + 6y = 14$

We can add the two equations to eliminate the variable $y$ .

Q: What is the graphical method?

The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. For example, if we have the equations:

$y = 2x - 1$

$y = 3x + 2$

We can graph the two equations on a coordinate plane and find the point of intersection.

Q: How do I know which method to use?

The choice of method depends on the type of equations and the variables involved. If the equations are linear and have two variables, the substitution or elimination method may be the best choice. If the equations are non-linear or have more than two variables, the graphical method may be the best choice.

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

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

Not checking the solution: Make sure to check the solution by substituting the values back into the original equations.
Not using the correct method: Choose the method that is best suited for the type of equations and variables involved.
Not simplifying the equations: Simplify the equations as much as possible to make it easier to solve.

Conclusion

Solving systems of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. By understanding the different methods for solving systems of equations and avoiding common mistakes, you can become proficient in solving these types of problems.

Final Answer

The final answer is $\boxed{0}$