Solve The Following System Of Equations.${ \begin{array}{l} -7x - 4y = 18 \ 3x - 2y = 22 \end{array} }$
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 can be represented graphically as a set of lines on a coordinate plane.
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 second equation for x:
3x - 2y = 22
Add 2y to both sides:
3x = 22 + 2y
Divide both sides by 3:
x = (22 + 2y) / 3
Step 2: Substitute the Expression into the Other Equation
Now, substitute the expression for x into the first equation:
-7x - 4y = 18
Substitute x = (22 + 2y) / 3:
-7((22 + 2y) / 3) - 4y = 18
Multiply both sides by 3 to eliminate the fraction:
-7(22 + 2y) - 12y = 54
Expand and simplify:
-154 - 14y - 12y = 54
Combine like terms:
-154 - 26y = 54
Add 154 to both sides:
-26y = 208
Divide both sides by -26:
y = -8
Step 3: Find the Value of the Other Variable
Now that we have the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the second equation:
3x - 2y = 22
Substitute y = -8:
3x - 2(-8) = 22
Simplify:
3x + 16 = 22
Subtract 16 from both sides:
3x = 6
Divide both sides by 3:
x = 2
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. Let's multiply the first equation by 3 and the second equation by 7:
-21x - 12y = 54
21x - 14y = 154
Step 2: Add or Subtract the Equations
Now, add the two equations to eliminate the variable x:
-12y + 21x - 14y = 54 + 154
Combine like terms:
7x - 26y = 208
Step 3: Solve for the Remaining Variable
Now that we have the equation 7x - 26y = 208, we can solve for y. Let's add 26y to both sides:
7x = 208 + 26y
Divide both sides by 7:
x = (208 + 26y) / 7
Now, substitute x = (208 + 26y) / 7 into one of the original equations to find the value of y. Let's use the first equation:
-7x - 4y = 18
Substitute x = (208 + 26y) / 7:
-7((208 + 26y) / 7) - 4y = 18
Simplify:
-208 - 26y - 4y = 18
Combine like terms:
-208 - 30y = 18
Add 208 to both sides:
-30y = 226
Divide both sides by -30:
y = -226 / 30
y = -7.53
Step 4: Find the Value of the Other Variable
Now that we have the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the second equation:
3x - 2y = 22
Substitute y = -7.53:
3x - 2(-7.53) = 22
Simplify:
3x + 15.06 = 22
Subtract 15.06 from both sides:
3x = 6.94
Divide both sides by 3:
x = 2.31
Conclusion
In this article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We found the values of x and y that satisfy both equations simultaneously. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one of the variables. Both methods are useful in solving systems of linear equations.
Real-World Applications
Solving systems of linear equations has numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, systems of linear equations can be used to model the motion of objects under the influence of forces. In engineering, systems of linear equations can be used to design and optimize systems such as electrical circuits and mechanical systems. In economics, systems of linear equations can be used to model the behavior of economic systems and make predictions about future trends.
Final Thoughts
Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. The method of substitution and elimination are two useful techniques for solving systems of linear equations. By understanding and applying these techniques, we can solve complex problems and make predictions about the behavior of systems.
Introduction
Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. However, it can be a challenging topic for many students and professionals. In this article, we will answer some frequently asked questions (FAQs) about solving 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 systems of linear equations?
A: There are two main methods of solving systems of linear equations: the method of substitution and the method of elimination. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one of the variables.
Q: What is the difference between the method of substitution and the method of elimination?
A: The main difference between the method of substitution and the method of elimination is the way in which the equations are manipulated. In the method of substitution, one equation is solved for one variable and then substituted into the other equation. In the method of elimination, the equations are added or subtracted to eliminate one of the variables.
Q: When should I use the method of substitution and when should I use the method of elimination?
A: You should use the method of substitution when one of the equations is easily solvable for one variable. You should use the method of elimination when the coefficients of one variable are the same in both equations.
Q: How do I know if a system of linear equations has a solution?
A: A system of linear equations has a solution if the lines represented by the equations intersect at a single point. If the lines are parallel, the system has no solution.
Q: How do I know if a system of linear equations has infinitely many solutions?
A: A system of linear equations has infinitely many solutions if the lines represented by the equations are identical. This occurs when the equations are equivalent.
Q: Can I use a calculator to solve systems of linear equations?
A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations.
Q: Can I use a computer program to solve systems of linear equations?
A: Yes, you can use a computer program to solve systems of linear equations. Many computer programs, such as MATLAB and Python, have built-in functions for solving systems of linear equations.
Q: What are some real-world applications of solving systems of linear equations?
A: Solving systems of linear equations has numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, systems of linear equations can be used to model the motion of objects under the influence of forces. In engineering, systems of linear equations can be used to design and optimize systems such as electrical circuits and mechanical systems.
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. By understanding and applying the methods of substitution and elimination, you can solve complex problems and make predictions about the behavior of systems. We hope that this article has answered some of your frequently asked questions about solving systems of linear equations.
Additional Resources
If you are interested in learning more about solving systems of linear equations, we recommend the following resources:
- Textbooks: "Linear Algebra and Its Applications" by Gilbert Strang and "Introduction to Linear Algebra" by Gilbert Strang
- Online Courses: "Linear Algebra" by MIT OpenCourseWare and "Linear Algebra" by Coursera
- Software: MATLAB and Python
- Websites: Khan Academy and Wolfram Alpha
We hope that this article has been helpful in answering your questions about solving systems of linear equations. If you have any further questions, please don't hesitate to ask.