Solve The System Of Equations Below.${ \begin{array}{l} 5x + 2y = 9 \ 2x - 3y = 15 \end{array} }$A. { (12, -3)$}$B. { (-3, 12)$}$C. { (-3, 3)$}$D. { (3, -3)$}$
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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 equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solution.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that involve two or more 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 system of equations is said to be consistent if it has a solution, and inconsistent if it does not have a solution.
The Method of Substitution
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the equations is easily solvable for one variable.
Step 1: Solve One Equation for One Variable
Let's solve the first equation for x:
5x + 2y = 9
Subtract 2y from both sides:
5x = 9 - 2y
Divide both sides by 5:
x = (9 - 2y) / 5
Step 2: Substitute the Expression into the Other Equation
Now, substitute the expression for x into the second equation:
2x - 3y = 15
Substitute x = (9 - 2y) / 5:
2((9 - 2y) / 5) - 3y = 15
Multiply both sides by 5 to eliminate the fraction:
2(9 - 2y) - 15y = 75
Expand and simplify:
18 - 4y - 15y = 75
Combine like terms:
-19y = 57
Divide both sides by -19:
y = -57 / 19
y = -3
Step 3: Find the Value of the Other Variable
Now that we have the value of y, substitute it back into one of the original equations to find the value of x. We will use the first equation:
5x + 2y = 9
Substitute y = -3:
5x + 2(-3) = 9
Simplify:
5x - 6 = 9
Add 6 to both sides:
5x = 15
Divide both sides by 5:
x = 15 / 5
x = 3
The Method of Elimination
The method of elimination involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of one variable are the same in both equations.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of that variable the same in both equations. We can do this by multiplying the equations by necessary multiples.
Multiply the first equation by 3 and the second equation by 2:
15x + 6y = 27
4x - 6y = 30
Step 2: Add or Subtract the Equations
Now, add the two equations to eliminate the variable y:
(15x + 6y) + (4x - 6y) = 27 + 30
Combine like terms:
19x = 57
Divide both sides by 19:
x = 57 / 19
x = 3
Step 3: Find the Value of the Other Variable
Now that we have the value of x, substitute it back into one of the original equations to find the value of y. We will use the first equation:
5x + 2y = 9
Substitute x = 3:
5(3) + 2y = 9
Simplify:
15 + 2y = 9
Subtract 15 from both sides:
2y = -6
Divide both sides by 2:
y = -6 / 2
y = -3
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have found the values of x and y that satisfy both equations simultaneously. The solution is x = 3 and y = -3.
Answer
The correct answer is:
A. (12, -3)
However, based on our calculations, the correct answer is:
(3, -3)
This is not among the options provided. It seems that there is an error in the question or the options provided.
Discussion
This problem is a classic example of a system of linear equations. It involves finding the values of two variables that satisfy two equations simultaneously. The method of substitution and elimination are two common methods used to solve such systems. In this article, we have used both methods to find the solution.
However, it seems that there is an error in the question or the options provided. The correct answer is (3, -3), which is not among the options provided. This highlights the importance of double-checking the question and the options provided before attempting to solve a problem.
Final Thoughts
Solving a system of linear equations is a fundamental concept in mathematics. It involves finding the values of variables that satisfy multiple equations simultaneously. The method of substitution and elimination are two common methods used to solve such systems. In this article, we have used both methods to find the solution. However, it seems that there is an error in the question or the options provided. The correct answer is (3, -3), which is not among the options provided.
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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 equations simultaneously. In this article, we will provide a Q&A guide to help you understand the concept and solve systems of linear equations.
Q: What is a System of Linear Equations?
A: A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: What are the Methods of Solving a System of Linear Equations?
A: There are two common methods of solving a system of linear equations:
- Method of Substitution: This method involves solving one equation for one variable and then substituting that expression into the other equation.
- Method of Elimination: This method involves adding or subtracting the equations to eliminate one of the variables.
Q: How Do I Choose Between the Two Methods?
A: You can choose between the two methods based on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can use the method of elimination. If the coefficients of one variable are different in both equations, you can use the method of substitution.
Q: What is the Difference Between a Consistent and Inconsistent System?
A: A consistent system is a system of linear equations that has a solution. An inconsistent system is a system of linear equations that does not have a solution.
Q: How Do I Determine Whether a System is Consistent or Inconsistent?
A: You can determine whether a system is consistent or inconsistent by checking 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 Solution to a System of Linear Equations?
A: The solution to a system of linear equations is the set of values of the variables that satisfy both equations simultaneously.
Q: How Do I Find the Solution to a System of Linear Equations?
A: You can find the solution to a system of linear equations by using the method of substitution or elimination. You can also use a graphing calculator or a computer program to find the solution.
Q: What are the Types of Solutions to a System of Linear Equations?
A: There are three types of solutions to a system of linear equations:
- Unique Solution: A system of linear equations has a unique solution if it has only one solution.
- Infinitely Many Solutions: A system of linear equations has infinitely many solutions if it has more than one solution.
- No Solution: A system of linear equations has no solution if it is inconsistent.
Q: How Do I Determine the Type of Solution to a System of Linear Equations?
A: You can determine the type of solution to a system of linear equations by checking if the equations are parallel or not. If the equations are parallel, the system has no solution. If the equations are not parallel, the system has a unique solution or infinitely many solutions.
Conclusion
Solving a system of linear equations is a fundamental concept in mathematics. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we have provided a Q&A guide to help you understand the concept and solve systems of linear equations. We hope this guide has been helpful in clarifying any doubts you may have had about solving systems of linear equations.
Final Thoughts
Solving a system of linear equations is a critical skill in mathematics and is used in a wide range of applications, including science, engineering, economics, and computer science. By understanding the concept and methods of solving systems of linear equations, you can solve a wide range of problems and make informed decisions in your personal and professional life.