Solve The Following System Of Equations:$\[ \begin{array}{l} 2x + 8y = 16 \\ -7y + 2x = -14 \end{array} \\]
===========================================================
Introduction
In mathematics, a system of linear equations is a set of two or more equations in which the variables are linear. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables.
The System of Equations
The given system of equations is:
{ \begin{array}{l} 2x + 8y = 16 \\ -7y + 2x = -14 \end{array} \}
Method 1: Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Let's solve the first equation for x:
2x = 16 - 8y
x = (16 - 8y) / 2
Now, substitute this expression for x into the second equation:
-7y + 2((16 - 8y) / 2) = -14
Simplify the equation:
-7y + 16 - 8y = -14
Combine like terms:
-15y + 16 = -14
Subtract 16 from both sides:
-15y = -30
Divide both sides by -15:
y = 2
Now that we have found 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:
2x + 8(2) = 16
Simplify the equation:
2x + 16 = 16
Subtract 16 from both sides:
2x = 0
Divide both sides by 2:
x = 0
Method 2: Elimination Method
The elimination method involves adding or subtracting the equations to eliminate one of the variables. Let's multiply the first equation by 7 and the second equation by 2 to make the coefficients of y opposites:
14x + 56y = 112
-14y + 4x = -28
Add the two equations together:
14x + 56y - 14y + 4x = 112 - 28
Combine like terms:
18x + 42y = 84
Now, let's multiply the first equation by 2 and the second equation by 7 to make the coefficients of x opposites:
4x + 16y = 32
-49y + 14x = -98
Add the two equations together:
4x + 16y - 49y + 14x = 32 - 98
Combine like terms:
18x - 33y = -66
Now we have two equations with the same coefficients for x:
18x + 42y = 84
18x - 33y = -66
Subtract the second equation from the first equation:
18x + 42y - (18x - 33y) = 84 - (-66)
Combine like terms:
75y = 150
Divide both sides by 75:
y = 2
Now that we have found 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:
2x + 8(2) = 16
Simplify the equation:
2x + 16 = 16
Subtract 16 from both sides:
2x = 0
Divide both sides by 2:
x = 0
Conclusion
In this article, we have solved a system of two linear equations with two variables using the substitution method and the elimination method. Both methods have led to the same solution: x = 0 and y = 2. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.
Tips and Tricks
- When solving a system of linear equations, it's essential to check the solution by substituting the values back into the original equations.
- The substitution method is often easier to use when one of the equations is already solved for one variable.
- The elimination method is often easier to use when the coefficients of one of the variables are opposites.
Real-World Applications
Solving systems of linear equations has numerous real-world applications, including:
- Physics and Engineering: Solving systems of linear equations is essential in physics and engineering to model real-world problems, such as the motion of objects or the behavior of electrical circuits.
- Economics: Solving systems of linear equations is used in economics to model economic systems, such as supply and demand curves.
- Computer Science: Solving systems of linear equations is used in computer science to solve problems in computer graphics, game development, and machine learning.
Final Thoughts
Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. By understanding the substitution method and the elimination method, you can solve systems of linear equations with ease and apply the concepts to real-world problems.
===========================================================
Q: What is a system of linear equations?
A system of linear equations is a set of two or more equations in which the variables are linear. In other words, it's a collection of equations where each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: How do I know which method to use to solve a system of linear equations?
The choice of method depends on the specific system of equations. If one of the equations is already solved for one variable, the substitution method may be easier to use. If the coefficients of one of the variables are opposites, the elimination method may be easier to use.
Q: What is the difference between the substitution method and the elimination method?
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables.
Q: How do I check my solution to a system of linear equations?
To check your solution, substitute the values back into the original equations. If the values satisfy both equations, then your solution is correct.
Q: What are some real-world applications of solving systems of linear equations?
Solving systems of linear equations has numerous real-world applications, including physics and engineering, economics, and computer science. It's used to model real-world problems, such as the motion of objects, supply and demand curves, and computer graphics.
Q: Can I use a calculator to solve a system of linear equations?
Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations.
Q: What if I have a system of linear equations with more than two variables?
If you have a system of linear equations with more than two variables, you can use the same methods as before, but you may need to use additional techniques, such as matrix operations or graphing.
Q: Can I use a computer program to solve a system of linear equations?
Yes, you can use a computer program to solve a system of linear equations. Many computer programs, such as MATLAB or Python, have built-in functions for solving systems of linear equations.
Q: What if I have a system of linear equations with no solution?
If you have a system of linear equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory, such as 2x + 3y = 5 and 2x + 3y = 7.
Q: What if I have a system of linear equations with infinitely many solutions?
If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent. This can happen if the equations are equivalent, such as 2x + 3y = 5 and 4x + 6y = 10.
Q: Can I use a graph to solve a system of linear equations?
Yes, you can use a graph to solve a system of linear equations. By graphing the equations on a coordinate plane, you can find the point of intersection, which represents the solution to the system.
Q: What are some common mistakes to avoid when solving systems of linear equations?
Some common mistakes to avoid when solving systems of linear equations include:
- Not checking the solution by substituting the values back into the original equations
- Not using the correct method for the specific system of equations
- Not simplifying the equations before solving
- Not using a calculator or computer program to check the solution
Q: How can I practice solving systems of linear equations?
You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also use a calculator or computer program to check your solutions and get feedback on your work.