Find The Solution Of The System Of Equations:$\[ \begin{aligned} 2x - 10y &= -28 \\ -10x + 10y &= -20 \end{aligned} \\]Answer Attempt 2 Out Of 3:( \[$\square\$\] , \[$\square\$\] )Submit Answer
**Solving Systems of Linear Equations: A Step-by-Step Guide** ===========================================================
Introduction
Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. 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 of the system.
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.
Example: Solving a System of Two Linear Equations
Let's consider the following system of two linear equations:
{ \begin{aligned} 2x - 10y &= -28 \\ -10x + 10y &= -20 \end{aligned} \}
Our goal is to find the values of x and y that satisfy both equations.
Method 1: Substitution Method
In the substitution method, we solve one equation for one variable and then substitute that expression into the other equation. Let's solve the first equation for x:
2x = 10y - 28
x = (10y - 28) / 2
Now, substitute this expression for x into the second equation:
-10((10y - 28) / 2) + 10y = -20
Simplify the equation:
-50y + 140 + 10y = -20
Combine like terms:
-40y + 140 = -20
Subtract 140 from both sides:
-40y = -160
Divide both sides by -40:
y = 4
Now that we have found 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:
2x - 10(4) = -28
Simplify the equation:
2x - 40 = -28
Add 40 to both sides:
2x = 12
Divide both sides by 2:
x = 6
Therefore, the solution to the system of equations is x = 6 and y = 4.
Method 2: Elimination Method
In the elimination method, we add or subtract the equations to eliminate one variable. Let's multiply the first equation by 5 and the second equation by 2 to make the coefficients of y opposites:
10x - 50y = -140
-20x + 20y = -40
Add both equations to eliminate y:
-10x - 30y = -180
Now, add 30y to both sides:
-10x = -180 + 30y
Divide both sides by -10:
x = (180 - 30y) / 10
Simplify the equation:
x = 18 - 3y
Now that we have found the expression for x, substitute it back into one of the original equations to find the value of y. Let's use the first equation:
2(18 - 3y) - 10y = -28
Simplify the equation:
36 - 6y - 10y = -28
Combine like terms:
36 - 16y = -28
Subtract 36 from both sides:
-16y = -64
Divide both sides by -16:
y = 4
Now that we have found the value of y, substitute it back into the expression for x:
x = 18 - 3(4)
Simplify the equation:
x = 18 - 12
x = 6
Therefore, the solution to the system of equations is x = 6 and y = 4.
Conclusion
Solving systems of linear equations is a crucial skill in mathematics and has numerous applications in various fields. In this article, we have used the substitution and elimination methods to solve a system of two linear equations with two variables. We have found that the solution to the system is x = 6 and y = 4.
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 involve two or more variables.
Q: What are the two methods to solve a system of linear equations?
A: The two methods to solve a system of linear equations are the substitution method and the elimination 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: How do I know if a system of linear equations has a solution?
A: A system of linear equations has a solution if it is consistent, and it does not have a solution if it is inconsistent.
Q: How do I find the solution of a system of linear equations?
A: To find the solution of a system of linear equations, you can use either the substitution method or the elimination method.
Q: What are the applications of solving systems of linear equations?
A: Solving systems of linear equations has numerous applications in various fields such as physics, engineering, economics, and computer science.
Q: How do I check if my solution is correct?
A: To check if your solution is correct, you can substitute the values of x and y back into the original equations to see if they are true.