Solve The Following System Of Equations:$\[ \begin{array}{l} 5x + 2y = 19 \\ 11x + Y = 18 \end{array} \\]
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation in the system is a linear equation, which means that it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are the variables.
Example: A System of Two Linear Equations
Consider the following system of two linear equations:
5x + 2y = 19
11x + y = 18
This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.
Method 1: Substitution Method
One way to solve a system of linear equations is to use the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve the First Equation for y
Let's solve the first equation for y:
5x + 2y = 19
Subtracting 5x from both sides gives:
2y = 19 - 5x
Dividing both sides by 2 gives:
y = (19 - 5x) / 2
Step 2: Substitute the Expression for y into the Second Equation
Now, let's substitute the expression for y into the second equation:
11x + y = 18
Substituting y = (19 - 5x) / 2 gives:
11x + (19 - 5x) / 2 = 18
Multiplying both sides by 2 gives:
22x + 19 - 5x = 36
Simplifying the equation gives:
17x = 17
Dividing both sides by 17 gives:
x = 1
Step 3: Find the Value of y
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
5x + 2y = 19
Substituting x = 1 gives:
5(1) + 2y = 19
Simplifying the equation gives:
5 + 2y = 19
Subtracting 5 from both sides gives:
2y = 14
Dividing both sides by 2 gives:
y = 7
Conclusion
In this article, we have solved a system of two linear equations with two variables using the substitution method. We have found the values of x and y that satisfy both equations, which are x = 1 and y = 7.
Method 2: Elimination Method
Another way to solve a system of linear equations is to use the elimination method. This method involves adding or subtracting the equations in the system to eliminate one of the variables.
Step 1: Multiply the Equations by Necessary Multiples
Let's multiply the first equation by -1 and the second equation by 2:
-1(5x + 2y = 19)
2(11x + y = 18)
This gives:
-5x - 2y = -19
22x + 2y = 36
Step 2: Add the Equations
Now, let's add the two equations:
(-5x - 2y) + (22x + 2y) = -19 + 36
Simplifying the equation gives:
17x = 17
Dividing both sides by 17 gives:
x = 1
Step 3: Find the Value of y
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
5x + 2y = 19
Substituting x = 1 gives:
5(1) + 2y = 19
Simplifying the equation gives:
5 + 2y = 19
Subtracting 5 from both sides gives:
2y = 14
Dividing both sides by 2 gives:
y = 7
Conclusion
In this article, we have solved a system of two linear equations with two variables using the elimination method. We have found the values of x and y that satisfy both equations, which are x = 1 and y = 7.
Method 3: Graphical Method
Another way to solve a system of linear equations is to use the graphical method. This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Step 1: Graph the Equations
Let's graph the two equations on a coordinate plane:
5x + 2y = 19
11x + y = 18
The graph of the first equation is a line with a slope of -5/2 and a y-intercept of 19/2. The graph of the second equation is a line with a slope of -11 and a y-intercept of 18.
Step 2: Find the Point of Intersection
The point of intersection of the two lines is the solution to the system of equations. We can find the point of intersection by finding the x-coordinate and the y-coordinate of the point where the two lines intersect.
x = 1
y = 7
Conclusion
In this article, we have solved a system of two linear equations with two variables using the graphical method. We have found the values of x and y that satisfy both equations, which are x = 1 and y = 7.
Conclusion
Introduction
In our previous article, we discussed how to solve a system of linear equations using three different methods: substitution, elimination, and graphical. In this article, we will answer some frequently asked questions about solving systems of linear equations.
Q: What is a system of linear equations?
A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation in the system is a linear equation, which means that it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are the variables.
Q: How do I know if a system of linear equations has a solution?
A system of linear equations has a solution if and only if the two equations are consistent. In other words, if the two equations have the same solution, then the system has a solution.
Q: What is the difference between a consistent and inconsistent system of linear equations?
A consistent system of linear equations is a system that has a solution. An inconsistent system of linear equations is a system that does not have a solution.
Q: How do I determine if a system of linear equations is consistent or inconsistent?
To determine if a system of linear equations is consistent or inconsistent, you can use the following methods:
- Substitution method: Substitute the expression for one variable into the other equation and simplify.
- Elimination method: Add or subtract the equations in the system to eliminate one of the variables.
- Graphical method: Graph the equations on a coordinate plane and find the point of intersection.
Q: What is the solution to a system of linear equations?
The solution to a system of linear equations is the set of values that satisfy both equations in the system. In other words, the solution is the point where the two lines intersect.
Q: How do I find the solution to a system of linear equations?
To find the solution to a system of linear equations, you can use the following methods:
- Substitution method: Substitute the expression for one variable into the other equation and simplify.
- Elimination method: Add or subtract the equations in the system to eliminate one of the variables.
- Graphical method: Graph the equations on a coordinate plane and find the point of intersection.
Q: What is the difference between a dependent and independent system of linear equations?
A dependent system of linear equations is a system that has infinitely many solutions. An independent system of linear equations is a system that has a unique solution.
Q: How do I determine if a system of linear equations is dependent or independent?
To determine if a system of linear equations is dependent or independent, you can use the following methods:
- Substitution method: Substitute the expression for one variable into the other equation and simplify.
- Elimination method: Add or subtract the equations in the system to eliminate one of the variables.
- Graphical method: Graph the equations on a coordinate plane and find the point of intersection.
Q: What is the significance of solving a system of linear equations?
Solving a system of linear equations is important in many fields, including mathematics, science, engineering, and economics. It is used to model real-world problems and to make predictions about the behavior of complex systems.
Conclusion
In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the definition of a system of linear equations, the difference between a consistent and inconsistent system, and the methods for solving a system of linear equations. We have also discussed the significance of solving a system of linear equations and the importance of understanding the concepts of dependent and independent systems.