Solve The System Of Equations:${ \begin{cases} 5x + 2y = 14 \ 3x - Y = 15 \end{cases} }$
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
Let's consider the following system of two linear equations:
{ \begin{cases} 5x + 2y = 14 \\ 3x - y = 15 \end{cases} \}
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 Second Equation for y
Let's solve the second equation for y:
3x - y = 15
Subtracting 3x from both sides gives:
-y = 15 - 3x
Multiplying both sides by -1 gives:
y = 3x - 15
Step 2: Substitute the Expression for y into the First Equation
Now, let's substitute the expression for y into the first equation:
5x + 2y = 14
Substituting y = 3x - 15 gives:
5x + 2(3x - 15) = 14
Expanding the equation gives:
5x + 6x - 30 = 14
Combine like terms:
11x - 30 = 14
Adding 30 to both sides gives:
11x = 44
Dividing both sides by 11 gives:
x = 4
Step 3: Find the Value of y
Now that we have found the value of x, we can find the value of y by substituting x = 4 into the expression for y:
y = 3x - 15
Substituting x = 4 gives:
y = 3(4) - 15
Expanding the equation gives:
y = 12 - 15
Simplifying the equation gives:
y = -3
Conclusion
We have solved the system of linear equations using the substitution method. The values of x and y that satisfy both equations are x = 4 and y = -3.
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 to eliminate one variable.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one variable, we need to multiply the equations by necessary multiples. Let's multiply the first equation by 1 and the second equation by 2:
First Equation: 5x + 2y = 14
Second Equation: 2(3x - y) = 2(15)
Expanding the second equation gives:
6x - 2y = 30
Step 2: Add the Equations
Now, let's add the equations to eliminate the variable y:
(5x + 2y) + (6x - 2y) = 14 + 30
Combine like terms:
11x = 44
Dividing both sides by 11 gives:
x = 4
Step 3: Find the Value of y
Now that we have found the value of x, we can find the value of y by substituting x = 4 into one of the original equations. Let's use the first equation:
5x + 2y = 14
Substituting x = 4 gives:
5(4) + 2y = 14
Expanding the equation gives:
20 + 2y = 14
Subtracting 20 from both sides gives:
2y = -6
Dividing both sides by 2 gives:
y = -3
Conclusion
We have solved the system of linear equations using the elimination method. The values of x and y that satisfy both equations are x = 4 and y = -3.
Conclusion
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. 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: 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. If the two equations are inconsistent, then the system has no 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 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 choose between the substitution method and the elimination method?
A: The choice of method depends on the specific problem and the variables involved. If one variable is already isolated in one of the equations, then the substitution method may be easier to use. If the coefficients of the variables are the same in both equations, then the elimination method may be 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 may need to use more complex techniques, such as matrix operations or Gaussian elimination.
Q: Can I use a calculator or computer to solve a system of linear equations?
A: Yes, you can use a calculator or computer to solve a system of linear equations. Many calculators and computer programs have built-in functions for solving systems of linear equations.
Q: What if I have a system of linear equations with no solution or infinitely many solutions?
A: If you have a system of linear equations with no solution, then the two equations are inconsistent, and there is no solution. If you have a system of linear equations with infinitely many solutions, then the two equations are dependent, and there are infinitely many solutions.
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 represents the solution to the system.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid when solving systems of linear equations include:
- Not checking for consistency before solving the system
- Not using the correct method for the problem
- Not checking for infinitely many solutions
- Not checking for no solution
Conclusion
In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the two main methods for solving systems of linear equations, the substitution method and the elimination method, and have provided some tips for choosing between the two methods. We have also discussed some common mistakes to avoid when solving systems of linear equations.