Solve The Following System Of Equations:${ \begin{align*} 3x + Y &= 5 \ 5x - 2y &= 1 \end{align*} }$
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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.
The System of Equations
The system of equations we will be solving is:
{ \begin{align*} 3x + y &= 5 \\ 5x - 2y &= 1 \end{align*} \}
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 this system is by using 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
We can start by solving the first equation for y:
3x + y = 5
Subtracting 3x from both sides gives us:
y = 5 - 3x
Step 2: Substitute the Expression for y into the Second Equation
Now that we have an expression for y, we can substitute it into the second equation:
5x - 2y = 1
Substituting y = 5 - 3x into this equation gives us:
5x - 2(5 - 3x) = 1
Expanding and simplifying this equation gives us:
5x - 10 + 6x = 1
Combine like terms:
11x - 10 = 1
Add 10 to both sides:
11x = 11
Divide both sides by 11:
x = 1
Step 3: Find the Value of y
Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. We will use the first equation:
3x + y = 5
Substituting x = 1 into this equation gives us:
3(1) + y = 5
Simplifying this equation gives us:
3 + y = 5
Subtracting 3 from both sides gives us:
y = 2
Method 2: Elimination Method
Another way to solve this system is by using the elimination method. This method involves adding or subtracting the equations in a way that eliminates one of the variables.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of y's in both equations are the same:
3x + y = 5
Multiplying this equation by 2 gives us:
6x + 2y = 10
5x - 2y = 1
Step 2: Add the Equations
Now that we have the equations with the same coefficients for y, we can add them to eliminate y:
(6x + 2y) + (5x - 2y) = 10 + 1
Simplifying this equation gives us:
11x = 11
Divide both sides by 11:
x = 1
Step 3: Find the Value of y
Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. We will use the first equation:
3x + y = 5
Substituting x = 1 into this equation gives us:
3(1) + y = 5
Simplifying this equation gives us:
3 + y = 5
Subtracting 3 from both sides gives us:
y = 2
Conclusion
In this article, we have solved a system of two linear equations with two variables using two different methods: substitution method and elimination method. Both methods have given us the same solution: x = 1 and y = 2. This solution satisfies both equations in the system, and it is the only solution that does so.
Real-World Applications
Solving systems of linear equations has many real-world applications. For example, in physics, we can use systems of linear equations to model the motion of objects. In economics, we can use systems of linear equations to model the behavior of markets. In engineering, we can use systems of linear equations to design and optimize systems.
Final Thoughts
Solving systems of linear equations is an important skill in mathematics and has many real-world applications. By using the substitution method and elimination method, we can solve systems of linear equations and find the values of the variables that satisfy all the equations in the system. With practice and experience, we can become proficient in solving systems of linear equations and apply this skill to a wide range of problems.
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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 the same set of variables. In other words, it is a collection of equations that can be written 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 if a system of linear equations has a solution?
A: A system of linear equations has a solution if and only if the two equations are consistent, meaning that they do not contradict each other. In other words, if the two equations have the same solution, then the system has a solution.
Q: What are the two main methods for solving systems of linear equations?
A: The two main methods for solving systems of linear equations are:
- 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 in a way that eliminates one of the variables.
Q: How do I choose between the substitution method and the elimination method?
A: The choice between the substitution method and the elimination method depends on the specific system of equations. If the coefficients of one variable are the same in both equations, then the elimination method is usually easier to use. If the coefficients of one variable are different in both equations, then the substitution method is usually easier to use.
Q: What if I have a system of linear equations with three or more variables?
A: If you have a system of linear equations with three or more variables, then you can use the same methods as before, but you will need to use more equations to eliminate the variables. For example, if you have a system of three linear equations with three variables, then you can use the elimination method to eliminate two of the variables and solve for the third variable.
Q: Can I use a calculator to solve a system of linear equations?
A: Yes, you can use a calculator to solve a system of linear equations. Most calculators have a built-in function for solving systems of linear equations, and you can enter the coefficients of the equations and the calculator will give you the solution.
Q: What if I have a system of linear equations with no solution?
A: If you have a system of linear equations with no solution, then it means that the two equations are inconsistent, meaning that they contradict each other. In this case, there is no value of the variables that can satisfy both equations.
Q: Can I use a graphing calculator to solve a system of linear equations?
A: Yes, you can use a graphing calculator to solve a system of linear equations. You can graph the two equations on the same coordinate plane and find the point of intersection, which is the solution to the system.
Q: What are some real-world applications of solving systems of linear equations?
A: Solving systems of linear equations has many real-world applications, including:
- Physics: Solving systems of linear equations can be used to model the motion of objects.
- Economics: Solving systems of linear equations can be used to model the behavior of markets.
- Engineering: Solving systems of linear equations can be used to design and optimize systems.
- Computer Science: Solving systems of linear equations can be used to solve problems in computer graphics and game development.
Q: How can I practice solving systems of linear equations?
A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own, using a calculator or graphing calculator to check your work.