For Each Ordered Pair, Determine Whether It Is A Solution To The System Of Equations.$\[ \begin{cases} 3x + 2y = -6 \\ 4x - 7y = -8 \end{cases} \\]\begin{tabular}{|c|c|c|}\hline\multirow{2}{*}{$(x, Y)$} & \multicolumn{2}{|c|}{Is It
Introduction
Systems of equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to determine whether a given ordered pair is a solution to a system of equations. We will use a step-by-step approach to solve the system of equations and then check if the given ordered pair satisfies the system.
What are Systems of Equations?
A system of equations is a set of two or more equations that contain the same variables. In this article, we will focus on a system of two linear equations in two variables. The general form of a linear equation is:
ax + by = c
where a, b, and c are constants, and x and y are the variables.
The System of Equations
The system of equations we will be working with is:
{ \begin{cases} 3x + 2y = -6 \\ 4x - 7y = -8 \end{cases} \}
This system consists of two linear equations in two variables, x and y.
Step 1: Write Down the Given Ordered Pair
The given ordered pair is (x, y). We will use this pair to check if it is a solution to the system of equations.
Step 2: Substitute the Ordered Pair into the First Equation
Substitute the ordered pair (x, y) into the first equation:
3x + 2y = -6
Replace x with the value from the ordered pair and simplify:
3(x) + 2(y) = -6
Step 3: Substitute the Ordered Pair into the Second Equation
Substitute the ordered pair (x, y) into the second equation:
4x - 7y = -8
Replace x with the value from the ordered pair and simplify:
4(x) - 7(y) = -8
Step 4: Check if the Ordered Pair Satisfies Both Equations
To check if the ordered pair satisfies both equations, we need to substitute the values of x and y into both equations and simplify. If the resulting equations are true, then the ordered pair is a solution to the system.
Let's assume the ordered pair is (x, y) = (2, -1). Substitute these values into both equations:
Equation 1:
3(2) + 2(-1) = -6
6 - 2 = -6
4 = -6
This equation is not true, so the ordered pair (2, -1) is not a solution to the system.
Equation 2:
4(2) - 7(-1) = -8
8 + 7 = -8
15 ≠-8
This equation is also not true, so the ordered pair (2, -1) is not a solution to the system.
Conclusion
In this article, we have explored how to determine whether a given ordered pair is a solution to a system of equations. We used a step-by-step approach to solve the system of equations and then checked if the given ordered pair satisfied the system. We found that the ordered pair (2, -1) is not a solution to the system.
Tips and Tricks
- When substituting the ordered pair into the equations, make sure to replace the variables with the correct values.
- Simplify the equations after substitution to check if they are true.
- If the ordered pair satisfies both equations, then it is a solution to the system.
Real-World Applications
Systems of equations have many real-world applications, such as:
- Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
- Economics: Systems of equations are used to model economic systems, such as supply and demand, and to make predictions about economic trends.
- Computer Science: Systems of equations are used in computer science to solve problems, such as linear programming and optimization.
Final Thoughts
Solving systems of equations is a crucial skill for students and professionals alike. By following the step-by-step approach outlined in this article, you can determine whether a given ordered pair is a solution to a system of equations. Remember to substitute the ordered pair into both equations, simplify, and check if the resulting equations are true. With practice and patience, you will become proficient in solving systems of equations and applying them to real-world problems.
Additional Resources
- Khan Academy: Systems of Equations
- Mathway: Systems of Equations
- Wolfram Alpha: Systems of Equations
Frequently Asked Questions
- Q: What is a system of equations? A: A system of equations is a set of two or more equations that contain the same variables.
- Q: How do I determine whether a given ordered pair is a solution to a system of equations? A: Substitute the ordered pair into both equations, simplify, and check if the resulting equations are true.
- Q: What are some real-world applications of systems of equations?
A: Systems of equations have many real-world applications, such as physics and engineering, economics, and computer science.
Frequently Asked Questions: Systems of Equations =====================================================
Q: What is a system of equations?
A: A system of equations is a set of two or more equations that contain the same variables. In this article, we will focus on a system of two linear equations in two variables.
Q: How do I determine whether a given ordered pair is a solution to a system of equations?
A: To determine whether a given ordered pair is a solution to a system of equations, you need to substitute the ordered pair into both equations, simplify, and check if the resulting equations are true.
Q: What are some common methods for solving systems of equations?
A: Some common methods for solving systems of equations include:
- Substitution Method: Substitute one equation into the other equation to solve for one variable.
- Elimination Method: Add or subtract the equations to eliminate one variable.
- Graphical Method: Graph the equations on a coordinate plane to find the point of intersection.
Q: What is the difference between a system of linear equations and a system of nonlinear equations?
A: A system of linear equations consists of two or more linear equations, while a system of nonlinear equations consists of two or more nonlinear equations. Nonlinear equations are equations that are not linear, such as quadratic or polynomial equations.
Q: Can a system of equations have no solution?
A: Yes, a system of equations can have no solution. This occurs when the equations are inconsistent, meaning that they cannot be true at the same time.
Q: Can a system of equations have an infinite number of solutions?
A: Yes, a system of equations can have an infinite number of solutions. This occurs when the equations are dependent, meaning that they are equivalent to each other.
Q: How do I know if a system of equations has a unique solution, no solution, or an infinite number of solutions?
A: To determine the number of solutions to a system of equations, you can use the following methods:
- Graphical Method: Graph the equations on a coordinate plane to see if they intersect at a single point, do not intersect, or are the same line.
- Substitution Method: Substitute one equation into the other equation to see if it is true or false.
- Elimination Method: Add or subtract the equations to see if it eliminates one variable.
Q: What are some real-world applications of systems of equations?
A: Systems of equations have many real-world applications, such as:
- Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
- Economics: Systems of equations are used to model economic systems, such as supply and demand, and to make predictions about economic trends.
- Computer Science: Systems of equations are used in computer science to solve problems, such as linear programming and optimization.
Q: How do I solve a system of equations with three or more variables?
A: To solve a system of equations with three or more variables, you can use the following methods:
- Substitution Method: Substitute one equation into another equation to solve for one variable.
- Elimination Method: Add or subtract the equations to eliminate one variable.
- Matrix Method: Use matrices to solve the system of equations.
Q: What are some common mistakes to avoid when solving systems of equations?
A: Some common mistakes to avoid when solving systems of equations include:
- Not checking if the equations are consistent or inconsistent.
- Not checking if the equations are dependent or independent.
- Not using the correct method to solve the system of equations.
Q: How do I check if a system of equations is consistent or inconsistent?
A: To check if a system of equations is consistent or inconsistent, you can use the following methods:
- Graphical Method: Graph the equations on a coordinate plane to see if they intersect at a single point or do not intersect.
- Substitution Method: Substitute one equation into the other equation to see if it is true or false.
- Elimination Method: Add or subtract the equations to see if it eliminates one variable.
Q: How do I check if a system of equations is dependent or independent?
A: To check if a system of equations is dependent or independent, you can use the following methods:
- Graphical Method: Graph the equations on a coordinate plane to see if they are the same line or intersect at a single point.
- Substitution Method: Substitute one equation into the other equation to see if it is true or false.
- Elimination Method: Add or subtract the equations to see if it eliminates one variable.
Additional Resources
- Khan Academy: Systems of Equations
- Mathway: Systems of Equations
- Wolfram Alpha: Systems of Equations
Frequently Asked Questions
- Q: What is a system of equations? A: A system of equations is a set of two or more equations that contain the same variables.
- Q: How do I determine whether a given ordered pair is a solution to a system of equations? A: Substitute the ordered pair into both equations, simplify, and check if the resulting equations are true.
- Q: What are some real-world applications of systems of equations? A: Systems of equations have many real-world applications, such as physics and engineering, economics, and computer science.