For Each Ordered Pair, Determine Whether It Is A Solution To The System Of Equations:$\[ \begin{cases} 4x - 5y = 8 \\ -3x + 2y = 1 \end{cases} \\]\begin{tabular}{|c|c|c|}\hline $(x, Y)$ & Yes & No \\\hline$(2, 0)$ & &

Feb 28, 2025 by ADMIN 226 views

Introduction to Systems of Equations

Systems of equations are a fundamental concept in mathematics, particularly in algebra. They consist of two or more equations that involve the same variables. In this article, we will focus on a system of two linear equations with two variables, x and y. The system is given by:

{ \begin{cases} 4x - 5y = 8 \\ -3x + 2y = 1 \end{cases} \}

Our goal is to determine whether a given ordered pair (x, y) is a solution to this system of equations. To do this, we will substitute the values of x and y into each equation and check if the resulting statement is true.

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. However, in this case, we will use the substitution method to check if a given ordered pair is a solution to the system.

Let's consider the ordered pair (2, 0). To determine whether it is a solution, we will substitute x = 2 and y = 0 into each equation.

Equation 1: 4x - 5y = 8

Substituting x = 2 and y = 0 into Equation 1, we get:

$4(2) - 5(0) = 8$

Simplifying the equation, we get:

$8 - 0 = 8$

This statement is true, so the ordered pair (2, 0) satisfies Equation 1.

Equation 2: -3x + 2y = 1

Substituting x = 2 and y = 0 into Equation 2, we get:

$-3(2) + 2(0) = 1$

Simplifying the equation, we get:

$-6 + 0 = 1$

This statement is false, so the ordered pair (2, 0) does not satisfy Equation 2.

Conclusion

Based on our analysis, we can conclude that the ordered pair (2, 0) is not a solution to the system of equations. This is because it does not satisfy both equations in the system.

Discussion

In this article, we used the substitution method to determine whether a given ordered pair is a solution to a system of equations. We substituted the values of x and y into each equation and checked if the resulting statement is true. This technique can be used to check if any ordered pair is a solution to a system of equations.

Example 1: Checking if (3, 2) is a Solution

Let's consider the ordered pair (3, 2). To determine whether it is a solution, we will substitute x = 3 and y = 2 into each equation.

Equation 1: 4x - 5y = 8

Substituting x = 3 and y = 2 into Equation 1, we get:

$4(3) - 5(2) = 8$

Simplifying the equation, we get:

$12 - 10 = 2$

This statement is false, so the ordered pair (3, 2) does not satisfy Equation 1.

Equation 2: -3x + 2y = 1

Substituting x = 3 and y = 2 into Equation 2, we get:

$-3(3) + 2(2) = 1$

Simplifying the equation, we get:

$-9 + 4 = -5$

This statement is false, so the ordered pair (3, 2) does not satisfy Equation 2.

Conclusion

Based on our analysis, we can conclude that the ordered pair (3, 2) is not a solution to the system of equations. This is because it does not satisfy both equations in the system.

Example 2: Checking if (0, 2) is a Solution

Let's consider the ordered pair (0, 2). To determine whether it is a solution, we will substitute x = 0 and y = 2 into each equation.

Equation 1: 4x - 5y = 8

Substituting x = 0 and y = 2 into Equation 1, we get:

$4(0) - 5(2) = 8$

Simplifying the equation, we get:

$0 - 10 = -10$

This statement is false, so the ordered pair (0, 2) does not satisfy Equation 1.

Equation 2: -3x + 2y = 1

Substituting x = 0 and y = 2 into Equation 2, we get:

$-3(0) + 2(2) = 1$

Simplifying the equation, we get:

$0 + 4 = 4$

This statement is false, so the ordered pair (0, 2) does not satisfy Equation 2.

Conclusion

Based on our analysis, we can conclude that the ordered pair (0, 2) is not a solution to the system of equations. This is because it does not satisfy both equations in the system.

Example 3: Checking if (4, 1) is a Solution

Let's consider the ordered pair (4, 1). To determine whether it is a solution, we will substitute x = 4 and y = 1 into each equation.

Equation 1: 4x - 5y = 8

Substituting x = 4 and y = 1 into Equation 1, we get:

$4(4) - 5(1) = 8$

Simplifying the equation, we get:

$16 - 5 = 11$

This statement is false, so the ordered pair (4, 1) does not satisfy Equation 1.

Equation 2: -3x + 2y = 1

Substituting x = 4 and y = 1 into Equation 2, we get:

$-3(4) + 2(1) = 1$

Simplifying the equation, we get:

$-12 + 2 = -10$

This statement is false, so the ordered pair (4, 1) does not satisfy Equation 2.

Conclusion

Based on our analysis, we can conclude that the ordered pair (4, 1) is not a solution to the system of equations. This is because it does not satisfy both equations in the system.

Example 4: Checking if (1, 3) is a Solution

Let's consider the ordered pair (1, 3). To determine whether it is a solution, we will substitute x = 1 and y = 3 into each equation.

Equation 1: 4x - 5y = 8

Substituting x = 1 and y = 3 into Equation 1, we get:

$4(1) - 5(3) = 8$

Simplifying the equation, we get:

$4 - 15 = -11$

This statement is false, so the ordered pair (1, 3) does not satisfy Equation 1.

Equation 2: -3x + 2y = 1

Substituting x = 1 and y = 3 into Equation 2, we get:

$-3(1) + 2(3) = 1$

Simplifying the equation, we get:

$-3 + 6 = 3$

This statement is false, so the ordered pair (1, 3) does not satisfy Equation 2.

Conclusion

Based on our analysis, we can conclude that the ordered pair (1, 3) is not a solution to the system of equations. This is because it does not satisfy both equations in the system.

Example 5: Checking if (2, 2) is a Solution

Let's consider the ordered pair (2, 2). To determine whether it is a solution, we will substitute x = 2 and y = 2 into each equation.

Equation 1: 4x - 5y = 8

Substituting x = 2 and y = 2 into Equation 1, we get:

$4(2) - 5(2) = 8$

Simplifying the equation, we get:

$8 - 10 = -2$

This statement is false, so the ordered pair (2, 2) does not satisfy Equation 1.

Equation 2: -3x + 2y = 1

Substituting x = 2 and y = 2 into Equation 2, we get:

$-3(2) + 2(2) = 1$

Simplifying the equation, we get:

$-6 + 4 = -2$

This statement is false, so the ordered pair (2, 2) does not satisfy Equation 2.

Conclusion

Based on our analysis, we can conclude that the ordered pair (2, 2) is not a solution to the system of equations. This is because it does not satisfy both equations in the system.

Example 6: Checking if (3, 3) is a Solution

Let's consider the ordered pair (3, 3). To determine whether it is a solution, we will substitute x = 3 and y = 3 into each equation.

Equation 1: 4x - 5y = 8

Substituting x = 3 and y = 3 into

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve the same variables. In this article, we will focus on a system of two linear equations with two variables, x and y.

Q: How do I determine whether an ordered pair is a solution to a system of equations?

A: To determine whether an ordered pair is a solution to a system of equations, you need to substitute the values of x and y into each equation and check if the resulting statement is true.

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. However, in this article, we will use the substitution method to check if a given ordered pair is a solution to the system.

Q: How do I know if an ordered pair is a solution to a system of equations?

A: An ordered pair is a solution to a system of equations if it satisfies both equations in the system. If it satisfies one equation but not the other, it is not a solution.

Q: What if I get a true statement when I substitute the values of x and y into one equation, but a false statement when I substitute them into the other equation?

A: If you get a true statement when you substitute the values of x and y into one equation, but a false statement when you substitute them into the other equation, then the ordered pair is not a solution to the system of equations.

Q: Can I use the substitution method to solve a system of equations?

A: Yes, you can use the substitution method to solve a system of equations. However, in this article, we will use the substitution method to check if a given ordered pair is a solution to the system.

Q: What if I get a true statement when I substitute the values of x and y into both equations?

A: If you get a true statement when you substitute the values of x and y into both equations, then the ordered pair is a solution to the system of equations.

Q: Can I use the substitution method to check if any ordered pair is a solution to a system of equations?

A: Yes, you can use the substitution method to check if any ordered pair is a solution to a system of equations.

Q: What if I get a false statement when I substitute the values of x and y into both equations?

A: If you get a false statement when you substitute the values of x and y into both equations, then the ordered pair is not a solution to the system of equations.

Q: Can I use the substitution method to solve a system of equations with more than two variables?

A: Yes, you can use the substitution method to solve a system of equations with more than two variables. However, in this article, we will focus on a system of two linear equations with two variables.

Q: What if I get a true statement when I substitute the values of x and y into one equation, but a false statement when I substitute them into the other equation, and then I get a true statement when I substitute the values of x and y into the other equation, but a false statement when I substitute them into the first equation?

Q: Can I use the substitution method to check if any ordered pair is a solution to a system of equations with more than two variables?

A: Yes, you can use the substitution method to check if any ordered pair is a solution to a system of equations with more than two variables.

Q: What if I get a false statement when I substitute the values of x and y into one equation, but a true statement when I substitute them into the other equation, and then I get a false statement when I substitute the values of x and y into the other equation, but a true statement when I substitute them into the first equation?

Q: Can I use the substitution method to solve a system of equations with more than two variables and more than two equations?

A: Yes, you can use the substitution method to solve a system of equations with more than two variables and more than two equations. However, in this article, we will focus on a system of two linear equations with two variables.

Q: What if I get a true statement when I substitute the values of x and y into one equation, but a false statement when I substitute them into the other equation, and then I get a true statement when I substitute the values of x and y into the other equation, but a false statement when I substitute them into the first equation, and then I get a true statement when I substitute the values of x and y into the first equation, but a false statement when I substitute them into the second equation?

A: If you get a true statement when you substitute the values of x and y into one equation, but a false statement when you substitute them into the other equation, and then you get a true statement when you substitute the values of x and y into the other equation, but a false statement when you substitute them into the first equation, and then you get a true statement when you substitute the values of x and y into the first equation, but a false statement when you substitute them into the second equation, then the ordered pair is not a solution to the system of equations.

Q: Can I use the substitution method to check if any ordered pair is a solution to a system of equations with more than two variables and more than two equations?

A: Yes, you can use the substitution method to check if any ordered pair is a solution to a system of equations with more than two variables and more than two equations.

Q: What if I get a false statement when I substitute the values of x and y into one equation, but a true statement when I substitute them into the other equation, and then I get a false statement when I substitute the values of x and y into the other equation, but a true statement when I substitute them into the first equation, and then I get a false statement when I substitute the values of x and y into the first equation, but a true statement when I substitute them into the second equation?

A: If you get a false statement when you substitute the values of x and y into one equation, but a true statement when you substitute them into the other equation, and then you get a false statement when you substitute the values of x and y into the other equation, but a true statement when you substitute them into the first equation, and then you get a false statement when you substitute the values of x and y into the first equation, but a true statement when you substitute them into the second equation, then the ordered pair is not a solution to the system of equations.

Q: Can I use the substitution method to solve a system of equations with more than two variables and more than two equations, and then use the substitution method to check if any ordered pair is a solution to the system?

A: Yes, you can use the substitution method to solve a system of equations with more than two variables and more than two equations, and then use the substitution method to check if any ordered pair is a solution to the system.

Q: What if I get a true statement when I substitute the values of x and y into one equation, but a false statement when I substitute them into the other equation, and then I get a true statement when I substitute the values of x and y into the other equation, but a false statement when I substitute them into the first equation, and then I get a true statement when I substitute the values of x and y into the first equation, but a false statement when I substitute them into the second equation, and then I get a true statement when I substitute the values of x and y into the second equation, but a false statement when I substitute them into the third equation?

A: If you get a true statement when you substitute the values of x and y into one equation, but a false statement when you substitute them into the other equation, and then you get a true statement when you substitute the values of x and y into the other equation, but a false statement when you substitute them into the first equation, and then you get a true statement when you substitute the values of x and y into the first equation, but a false statement when you substitute them into the second equation, and then you get a true statement when you substitute the values of x and y into the second equation, but a false statement when you substitute them into the third equation, then the ordered pair is not a solution to the system of equations.

Q: Can I use the substitution method to check if any ordered pair is a solution to a system of equations with more than two variables and more than two equations, and then use the substitution method to solve the system?

A: Yes, you can use the substitution method to check if any ordered pair is a solution to a system of equations with more than two variables and more than two equations, and then use the substitution method to solve the system.