Find The Solution Of The System Of Equations.$\[ \begin{aligned} -7x - 4y &= -44 \\ 7x - 3y &= 16 \end{aligned} \\]
**Solving a System of Linear Equations: A Step-by-Step Guide** ===========================================================
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of two linear equations with two variables.
The System of Equations
The system of equations we will be solving is:
{ \begin{aligned} -7x - 4y &= -44 \\ 7x - 3y &= 16 \end{aligned} \}
Why Do We Need to Solve a System of Equations?
Solving a system of equations is essential in various fields such as physics, engineering, economics, and computer science. It helps us to model real-world problems and find the values of the variables that satisfy the given conditions.
How to Solve a System of Equations?
There are several methods to solve a system 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 variable.
- Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Step-by-Step Solution Using the Substitution Method
Step 1: Solve One Equation for One Variable
Let's solve the first equation for x:
{ -7x - 4y = -44 \}
Add 4y to both sides:
{ -7x = -44 + 4y \}
Divide both sides by -7:
{ x = \frac{-44 + 4y}{-7} \}
Step 2: Substitute the Expression into the Other Equation
Now, substitute the expression for x into the second equation:
{ 7\left(\frac{-44 + 4y}{-7}\right) - 3y = 16 \}
Simplify the equation:
{ -44 + 4y - 3y = 16 \}
Combine like terms:
{ -44 + y = 16 \}
Step 3: Solve for the Variable
Add 44 to both sides:
{ y = 16 + 44 \}
Simplify:
{ y = 60 \}
Step 4: 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. Let's use the first equation:
{ -7x - 4(60) = -44 \}
Simplify:
{ -7x - 240 = -44 \}
Add 240 to both sides:
{ -7x = 196 \}
Divide both sides by -7:
{ x = \frac{196}{-7} \}
Simplify:
{ x = -28 \}
Step-by-Step Solution Using the Elimination Method
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one variable, we need to multiply the equations by necessary multiples such that the coefficients of y's in both equations are the same. Let's multiply the first equation by 3 and the second equation by 4:
{ -21x - 12y = -132 \}
{ 28x - 12y = 64 \}
Step 2: Add or Subtract the Equations
Now, add the two equations to eliminate the y variable:
{ -21x - 12y + 28x - 12y = -132 + 64 \}
Simplify:
{ 7x - 24y = -68 \}
Step 3: Solve for the Variable
Now that we have the equation with only one variable, solve for x:
{ 7x = -68 + 24y \}
Divide both sides by 7:
{ x = \frac{-68 + 24y}{7} \}
Step 4: 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. Let's use the first equation:
{ -7\left(\frac{-68 + 24y}{7}\right) - 4y = -44 \}
Simplify:
{ -68 + 24y - 4y = -44 \}
Combine like terms:
{ -68 + 20y = -44 \}
Add 68 to both sides:
{ 20y = 24 \}
Divide both sides by 20:
{ y = \frac{24}{20} \}
Simplify:
{ y = \frac{6}{5} \}
Step 5: 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. Let's use the first equation:
{ -7x - 4\left(\frac{6}{5}\right) = -44 \}
Simplify:
{ -7x - \frac{24}{5} = -44 \}
Multiply both sides by 5:
{ -35x - 24 = -220 \}
Add 24 to both sides:
{ -35x = -196 \}
Divide both sides by -35:
{ x = \frac{-196}{-35} \}
Simplify:
{ x = \frac{56}{35} \}
Conclusion
In this article, we have learned how to solve a system of linear equations using the substitution method and the elimination method. We have also seen how to find the values of the variables using these methods. By following the steps outlined in this article, you can solve any system of linear equations.
Frequently Asked Questions
Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.
Q: Why do we need to solve a system of equations?
A: Solving a system of equations is essential in various fields such as physics, engineering, economics, and computer science. It helps us to model real-world problems and find the values of the variables that satisfy the given conditions.
Q: What are the methods to solve a system of equations?
A: There are several methods to solve a system of equations, including the substitution method, the elimination method, and the graphical method.
Q: What is the substitution method?
A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
Q: What is the elimination method?
A: The elimination method involves adding or subtracting the equations to eliminate one variable.
Q: What is the graphical method?
A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: How do I choose the method to solve a system of equations?
A: The choice of method depends on the type of equations and the variables involved. The substitution method is often used when one equation is easily solvable for one variable, while the elimination method is often used when the coefficients of the variables are the same.
Q: What are the steps to solve a system of equations using the substitution method?
A: The steps to solve a system of equations using the substitution method are:
- Solve one equation for one variable.
- Substitute the expression into the other equation.
- Solve for the variable.
- Find the value of the other variable.
Q: What are the steps to solve a system of equations using the elimination method?
A: The steps to solve a system of equations using the elimination method are:
- Multiply the equations by necessary multiples.
- Add or subtract the equations to eliminate one variable.
- Solve for the variable.
- Find the value of the other variable.
Q: What are the advantages of solving a system of equations?
A: The advantages of solving a system of equations include:
- Modeling real-world problems
- Finding the values of the variables that satisfy the given conditions
- Understanding the relationships between the variables
Q: What are the disadvantages of solving a system of equations?
A: The disadvantages of solving a system of equations include:
- Complexity of the equations
- Difficulty in finding the values of the variables
- Limited applicability of the methods
References
- [1] "Solving Systems of Linear Equations" by Math Is Fun
- [2] "Systems of Linear Equations" by Khan Academy
- [3] "Solving Systems of Linear Equations" by Purplemath
Additional Resources
- [1] "Solving Systems of Linear Equations" by MIT OpenCourseWare
- [2] "Systems of Linear Equations" by Wolfram Alpha
- [3] "Solving Systems of Linear Equations" by Mathway