Solve The System Of Equations:${ \begin{align*} x + 3y &= 9 \ 4x - 2y &= -6 \end{align*} }$
Introduction
Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. 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.
What are Systems 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 goal is to find the values of x and y that satisfy all the equations in the system.
The Method of Substitution
One way to solve a system of linear equations is by using the method of substitution. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve One Equation for One Variable
Let's start by solving the first equation for x:
x + 3y = 9
Subtract 3y from both sides:
x = 9 - 3y
Step 2: Substitute the Expression into the Other Equation
Now, substitute the expression for x into the second equation:
4(9 - 3y) - 2y = -6
Expand and simplify:
36 - 12y - 2y = -6
Combine like terms:
-14y = -42
Divide both sides by -14:
y = 3
Step 3: Find the Value of x
Now that we have 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:
x + 3y = 9
Substitute y = 3:
x + 3(3) = 9
Simplify:
x + 9 = 9
Subtract 9 from both sides:
x = 0
The Method of Elimination
Another way to solve a system of linear equations is by using the method of elimination. This method involves adding or subtracting the equations to eliminate 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 the variable to be eliminated are the same.
Multiply the first equation by 2 and the second equation by 3:
2(x + 3y) = 2(9) 3(4x - 2y) = 3(-6)
Simplify:
2x + 6y = 18 12x - 6y = -18
Step 2: Add or Subtract the Equations
Now, add the two equations to eliminate the variable y:
(2x + 6y) + (12x - 6y) = 18 + (-18)
Combine like terms:
14x = 0
Divide both sides by 14:
x = 0
Step 3: Find the Value of y
Now that we have the value of x, substitute it back into one of the original equations to find the value of y. We will use the first equation:
x + 3y = 9
Substitute x = 0:
0 + 3y = 9
Simplify:
3y = 9
Divide both sides by 3:
y = 3
Conclusion
In this article, we have discussed two methods for solving systems of linear equations: the method of substitution and the method of elimination. We have used these methods to solve a system of two linear equations with two variables. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one of the variables. Both methods are useful tools for solving systems of linear equations.
Example Problems
Problem 1
Solve the system of linear equations:
x + 2y = 6 3x - 4y = -2
Solution
Using the method of substitution, we can solve this system of linear equations as follows:
x + 2y = 6
Subtract 2y from both sides:
x = 6 - 2y
Substitute this expression into the second equation:
3(6 - 2y) - 4y = -2
Expand and simplify:
18 - 6y - 4y = -2
Combine like terms:
-10y = -20
Divide both sides by -10:
y = 2
Substitute y = 2 back into one of the original equations to find the value of x. We will use the first equation:
x + 2y = 6
Substitute y = 2:
x + 2(2) = 6
Simplify:
x + 4 = 6
Subtract 4 from both sides:
x = 2
Problem 2
Solve the system of linear equations:
2x + 3y = 7 x - 2y = -3
Solution
Using the method of elimination, we can solve this system of linear equations as follows:
Multiply the first equation by 2 and the second equation by 3:
2(2x + 3y) = 2(7) 3(x - 2y) = 3(-3)
Simplify:
4x + 6y = 14 3x - 6y = -9
Add the two equations to eliminate the variable y:
(4x + 6y) + (3x - 6y) = 14 + (-9)
Combine like terms:
7x = 5
Divide both sides by 7:
x = 5/7
Substitute x = 5/7 back into one of the original equations to find the value of y. We will use the first equation:
2x + 3y = 7
Substitute x = 5/7:
2(5/7) + 3y = 7
Simplify:
10/7 + 3y = 7
Multiply both sides by 7:
10 + 21y = 49
Subtract 10 from both sides:
21y = 39
Divide both sides by 21:
y = 39/21
Simplify:
y = 13/7
Final Answer
Introduction
In our previous article, we discussed the methods of substitution and elimination for solving systems of linear equations. 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 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.
Q: How do I know which method to use?
The choice of method depends on the coefficients of the variables in the equations. If the coefficients are the same, use the method of elimination. If the coefficients are different, use the method of substitution.
Q: What is the difference between the method of substitution and the method of elimination?
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one of the variables.
Q: How do I solve a system of linear equations with three variables?
To solve a system of linear equations with three variables, you can use the method of substitution or elimination. However, it is often easier to use the method of substitution. First, solve one equation for one variable. Then, substitute that expression into the other two equations. Finally, solve the resulting system of two linear equations.
Q: What if I have a system of linear equations with more than three variables?
If you have a system of linear equations with more than three variables, you can use the method of substitution or elimination. However, it may be more difficult to solve the system. In this case, you may need to use a computer or calculator to solve the system.
Q: Can I use a graphing calculator to solve a system of linear equations?
Yes, you can use a graphing calculator to solve a system of linear equations. Graphing calculators can be used to graph the equations and find the point of intersection, which is the solution to the system.
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 = 10.
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 identical, such as 2x + 3y = 5 and 2x + 3y = 5.
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. Computer programs such as MATLAB, Python, and R can be used to solve systems of linear equations.
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 if the equations are consistent or inconsistent
- Not checking if the equations are dependent or independent
- Not using the correct method for solving the system
- Not checking if the solution is unique or not
Conclusion
Solving systems of linear equations is an important skill in mathematics and science. By understanding the methods of substitution and elimination, you can solve systems of linear equations with ease. Remember to check if the equations are consistent or inconsistent, and to use the correct method for solving the system. With practice and patience, you can become proficient in solving systems of linear equations.
Example Problems
Problem 1
Solve the system of linear equations:
x + 2y = 6 3x - 4y = -2
Solution
Using the method of substitution, we can solve this system of linear equations as follows:
x + 2y = 6
Subtract 2y from both sides:
x = 6 - 2y
Substitute this expression into the second equation:
3(6 - 2y) - 4y = -2
Expand and simplify:
18 - 6y - 4y = -2
Combine like terms:
-10y = -20
Divide both sides by -10:
y = 2
Substitute y = 2 back into one of the original equations to find the value of x. We will use the first equation:
x + 2y = 6
Substitute y = 2:
x + 2(2) = 6
Simplify:
x + 4 = 6
Subtract 4 from both sides:
x = 2
Problem 2
Solve the system of linear equations:
2x + 3y = 7 x - 2y = -3
Solution
Using the method of elimination, we can solve this system of linear equations as follows:
Multiply the first equation by 2 and the second equation by 3:
2(2x + 3y) = 2(7) 3(x - 2y) = 3(-3)
Simplify:
4x + 6y = 14 3x - 6y = -9
Add the two equations to eliminate the variable y:
(4x + 6y) + (3x - 6y) = 14 + (-9)
Combine like terms:
7x = 5
Divide both sides by 7:
x = 5/7
Substitute x = 5/7 back into one of the original equations to find the value of y. We will use the first equation:
2x + 3y = 7
Substitute x = 5/7:
2(5/7) + 3y = 7
Simplify:
10/7 + 3y = 7
Multiply both sides by 7:
10 + 21y = 49
Subtract 10 from both sides:
21y = 39
Divide both sides by 21:
y = 39/21
Simplify:
y = 13/7
Final Answer
The final answer is x = 2 and y = 2.