What Is The Solution Of The System Of Equations?$\[ \left\{ \begin{array}{l} \frac{5}{3} X - 2 Y = -2 \\ -x + Y = 2 \end{array} \right. \\]Enter Your Answer In The Boxes.
Introduction
In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. Solving a system of equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will discuss how to solve a system of equations using the method of substitution and elimination.
What is a System of Equations?
A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. Each equation in the system is a statement that two expressions are equal. For example, the system of equations:
{ \left\{ \begin{array}{l} \frac{5}{3} x - 2 y = -2 \\ -x + y = 2 \end{array} \right. \}
is a system of two equations with two variables, x and y.
Types of Systems of Equations
There are two types of systems of equations: linear and nonlinear. A linear system of equations is a system of equations in which each equation is a linear equation, i.e., an equation in which the highest power of the variables is 1. A nonlinear system of equations is a system of equations in which at least one equation is a nonlinear equation, i.e., an equation in which the highest power of the variables is greater than 1.
Solving a System of Equations
There are several methods for solving a system of equations, including the method of substitution, the method of elimination, and the method of matrices. In this article, we will discuss the method of substitution and elimination.
Method of Substitution
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. For example, to solve the system of equations:
{ \left\{ \begin{array}{l} \frac{5}{3} x - 2 y = -2 \\ -x + y = 2 \end{array} \right. \}
we can solve the second equation for y:
y = x + 2
and then substitute that expression into the first equation:
$\frac{5}{3} x - 2 (x + 2) = -2$
Simplifying the equation, we get:
$\frac{5}{3} x - 2x - 4 = -2$
Combine like terms:
$\frac{5}{3} x - 2x = 2$
$\frac{5}{3} x - \frac{6}{3} x = 2$
$-\frac{1}{3} x = 2$
Multiply both sides by -3:
$x = -6$
Now that we have found the value of x, we can substitute that value into one of the original equations to find the value of y. Substituting x = -6 into the second equation, we get:
$-(-6) + y = 2$
Simplifying the equation, we get:
$6 + y = 2$
Subtract 6 from both sides:
$y = -4$
Therefore, the solution to the system of equations is x = -6 and y = -4.
Method of Elimination
The method of elimination involves adding or subtracting the equations in the system to eliminate one of the variables. For example, to solve the system of equations:
{ \left\{ \begin{array}{l} \frac{5}{3} x - 2 y = -2 \\ -x + y = 2 \end{array} \right. \}
we can add the two equations to eliminate the variable y:
$\frac{5}{3} x - 2 y + (-x + y) = -2 + 2$
Simplifying the equation, we get:
$\frac{5}{3} x - x - 2 y + y = 0$
Combine like terms:
$\frac{5}{3} x - x - y = 0$
$\frac{2}{3} x - y = 0$
Add y to both sides:
$\frac{2}{3} x = y$
Now that we have eliminated the variable y, we can solve for x. Multiply both sides by 3/2:
$x = \frac{3}{2} y$
Substitute this expression for x into one of the original equations to find the value of y. Substituting x = (3/2)y into the second equation, we get:
$-(\frac{3}{2} y) + y = 2$
Simplifying the equation, we get:
$-\frac{3}{2} y + y = 2$
Combine like terms:
$-\frac{1}{2} y = 2$
Multiply both sides by -2:
$y = -4$
Now that we have found the value of y, we can substitute that value into one of the original equations to find the value of x. Substituting y = -4 into the expression x = (3/2)y, we get:
$x = \frac{3}{2} (-4)$
Simplifying the equation, we get:
$x = -6$
Therefore, the solution to the system of equations is x = -6 and y = -4.
Conclusion
In this article, we discussed how to solve a system of equations using the method of substitution and elimination. We also discussed the types of systems of equations and the methods for solving them. 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 in the system to eliminate one of the variables. By following these methods, we can solve systems of equations and find the values of the variables.
Example Problems
Problem 1
Solve the system of equations:
{ \left\{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \right. \}
Solution
To solve the system of equations, we can use the method of substitution. We can solve the second equation for x:
x = -3 + 2y
and then substitute that expression into the first equation:
2(-3 + 2y) + 3y = 7
Simplifying the equation, we get:
-6 + 4y + 3y = 7
Combine like terms:
7y = 13
Divide both sides by 7:
y = \frac{13}{7}
Now that we have found the value of y, we can substitute that value into one of the original equations to find the value of x. Substituting y = (13/7) into the expression x = (-3 + 2y), we get:
x = -3 + 2(\frac{13}{7})
Simplifying the equation, we get:
x = -3 + \frac{26}{7}
x = \frac{-21 + 26}{7}
x = \frac{5}{7}
Therefore, the solution to the system of equations is x = (5/7) and y = (13/7).
Problem 2
Solve the system of equations:
{ \left\{ \begin{array}{l} x + 2y = 4 \\ 3x - 2y = 5 \end{array} \right. \}
Solution
To solve the system of equations, we can use the method of elimination. We can add the two equations to eliminate the variable y:
x + 2y + 3x - 2y = 4 + 5
Simplifying the equation, we get:
4x = 9
Divide both sides by 4:
x = \frac{9}{4}
Now that we have found the value of x, we can substitute that value into one of the original equations to find the value of y. Substituting x = (9/4) into the first equation, we get:
(\frac{9}{4}) + 2y = 4
Simplifying the equation, we get:
2y = 4 - \frac{9}{4}
2y = \frac{16 - 9}{4}
2y = \frac{7}{4}
Divide both sides by 2:
y = \frac{7}{8}
Therefore, the solution to the system of equations is x = (9/4) and y = (7/8).
Final Answer
Q: What is a system of equations?
A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
Q: What are the types of systems of equations?
A: There are two types of systems of equations: linear and nonlinear. A linear system of equations is a system of equations in which each equation is a linear equation, i.e., an equation in which the highest power of the variables is 1. A nonlinear system of equations is a system of equations in which at least one equation is a nonlinear equation, i.e., an equation in which the highest power of the variables is greater than 1.
Q: How do I solve a system of equations?
A: There are several methods for solving a system of equations, including the method of substitution, the method of elimination, and the method of matrices. 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 in the system to eliminate one of the variables.
Q: What is the method of substitution?
A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. For example, to solve the system of equations:
{ \left\{ \begin{array}{l} \frac{5}{3} x - 2 y = -2 \\ -x + y = 2 \end{array} \right. \}
we can solve the second equation for y:
y = x + 2
and then substitute that expression into the first equation:
$\frac{5}{3} x - 2 (x + 2) = -2$
Q: What is the method of elimination?
A: The method of elimination involves adding or subtracting the equations in the system to eliminate one of the variables. For example, to solve the system of equations:
{ \left\{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \right. \}
we can add the two equations to eliminate the variable y:
2x + 3y + x - 2y = 7 + (-3)
Q: How do I know which method to use?
A: The choice of method depends on the specific system of equations and the variables involved. If the system of equations has two variables and two equations, the method of substitution or elimination may be used. If the system of equations has more than two variables or more than two equations, the method of matrices may be used.
Q: What is the method of matrices?
A: The method of matrices involves representing the system of equations as a matrix and then using row operations to solve for the variables. This method is often used for systems of equations with more than two variables or more than two equations.
Q: How do I represent a system of equations as a matrix?
A: To represent a system of equations as a matrix, we can write the coefficients of the variables in the equations as a matrix, with the variables as the columns. For example, the system of equations:
{ \left\{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \right. \}
can be represented as the matrix:
$\begin{bmatrix} 2 & 3 \\ 1 & -2 \end{bmatrix}$
Q: How do I use row operations to solve a system of equations?
A: To use row operations to solve a system of equations, we can perform row operations on the matrix to transform it into a matrix with a solution. For example, to solve the system of equations:
{ \left\{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \right. \}
we can perform the following row operations:
$\begin{bmatrix} 2 & 3 \\ 1 & -2 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix}$
$\begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & -2 \\ 0 & 7 \end{bmatrix}$
$\begin{bmatrix} 1 & -2 \\ 0 & 7 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 7 \end{bmatrix}$
Q: What is the solution to the system of equations?
A: The solution to the system of equations is the values of the variables that satisfy all the equations in the system. In the case of the system of equations:
{ \left\{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \right. \}
the solution is x = (9/4) and y = (7/8).
Q: How do I check my solution?
A: To check your solution, you can substitute the values of the variables into one of the original equations and verify that the equation is satisfied. For example, to check the solution x = (9/4) and y = (7/8) to the system of equations:
{ \left\{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \right. \}
we can substitute x = (9/4) and y = (7/8) into the first equation:
2(\frac{9}{4}) + 3(\frac{7}{8}) = 7
Simplifying the equation, we get:
$\frac{18}{4} + \frac{21}{8} = 7$
$\frac{36}{8} + \frac{21}{8} = 7$
$\frac{57}{8} = 7$
Since the equation is not satisfied, the solution x = (9/4) and y = (7/8) is not correct.
Q: What if I get a solution that doesn't satisfy the equations?
A: If you get a solution that doesn't satisfy the equations, it means that the solution is not correct. You should go back and recheck your work to see where you made the mistake.