Introduction
System of equations is a set of two or more equations that must be solved simultaneously to find the values of the variables in each equation. This is a fundamental problem in algebra and is important in many branches of science and engineering. There are several methods to solve systems of equations, and in this article, we’ll discuss five different methods: graphical, substitution, elimination, matrix, and Cramer’s rule. These methods are widely used in real-world problems and can help you understand the concepts and master the skill of solving system of equations with ease.
Step-wise approach to solve a system of equations
The general process to solve a system of equations involves the following steps:
- Identify the variables in each equation, and write them in a vertical column.
- Write the coefficients of each variable in each equation in a matrix form.
- Use one of the methods discussed below to solve the system of equations.
- Check the validity of the solution by substituting it in each equation.
Let’s use the following system of equations to illustrate the process:
4x + 3y = 8
2x – 5y = -7
We can write the variables in a column as:
x
y
And the coefficients of each variable in each equation in matrix form as:
[4 3 | 8]
[2 -5 |-7]
Graphical method to solve a system of equations
The graphical method involves plotting the equations on a graph and finding the point where they intersect. This method is useful to solve a system of equations with two variables.
Let’s use the same system of equations from above to illustrate the graphical method:
4x + 3y = 8
2x – 5y = -7
First, we’ll rearrange the equations in slope-intercept form:
y = (-4/3)x + (8/3)
y = (2/5)x + (7/5)
Then, we’ll plot these two lines on a graph:
The point where the two lines intersect is the solution to the system of equations, which is approximately (x, y) = (1.14, 1.71).
The advantages of using the graphical method are that it’s easy to understand, and it provides a visual representation of the solution. However, this method can be inaccurate, especially when the solution is not an integer value.
Substitution method to solve a system of equations
The substitution method involves solving one of the equations for one of the variables and substituting that expression into the other equation. This method is useful when one of the equations has a variable expressed in terms of the other variable.
Let’s use the same system of equations from above to illustrate the substitution method:
4x + 3y = 8
2x – 5y = -7
We can solve the second equation for x as:
x = (5y-7)/2
Then, we substitute this expression for x into the first equation:
4((5y-7)/2) + 3y = 8
Simplifying the expression, we get:
8y – 14 + 3y = 8
11y = 22
y = 2
Now, we substitute this value for y into the first equation:
4x + 3(2) = 8
4x + 6 = 8
4x = 2
x = 0.5
Therefore, the solution to the system of equations is (x, y) = (0.5, 2).
The advantages of using the substitution method are that it’s straightforward and easy to understand. However, this method can be time-consuming, especially when one of the equations has a complicated expression for one of the variables.
Elimination method to solve a system of equations
The elimination method involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of one of the variables are opposites.
Let’s use the same system of equations from above to illustrate the elimination method:
4x + 3y = 8
2x – 5y = -7
We can multiply the first equation by 5 and the second equation by 3 to eliminate y:
20x + 15y = 40
6x – 15y = -21
Adding these two equations, we can eliminate y:
26x = 19
x = 19/26
Now, we substitute this value for x into one of the original equations:
4(19/26) + 3y = 8
Simplifying the equation, we get:
3y = 34/13
y = 34/39
Therefore, the solution to the system of equations is (x, y) = (19/26, 34/39).
The advantages of using the elimination method are that it’s quick and efficient when the coefficients are opposites. However, this method requires some skill in manipulating equations.
Matrix method to solve a system of equations
The matrix method involves using matrices to represent the system of equations. This method is useful when the number of variables is larger than two.
Let’s use the following system of equations to illustrate the matrix method:
2x + 3y – z = 1
x + y + z = 6
2x – y + z = 1
We can represent the coefficients of each variable in each equation in matrix form as:
[2 3 -1 | 1]
[1 1 1 | 6]
[2 -1 1 | 1]
Then, we place the matrix in row echelon form by performing elementary row operations:
[1 1 1 | 6]
[0 5 -3 | 5]
[0 0 1 | -4]
Now, we can read the solutions directly from the matrix:
z = -4
y = 1
x = 2
Therefore, the solution to the system of equations is (x, y, z) = (2, 1, -4).
The advantages of using the matrix method are that it’s efficient and can be used for systems with any number of variables. However, this method may require some skill in matrix manipulation.
Cramer’s rule to solve a system of equations
Cramer’s rule involves using determinants to solve a system of equations. This method is useful when the number of variables is equal to the number of equations.
Let’s use the following system of equations to illustrate Cramer’s rule:
2x + 3y = 8
4x – 5y = -7
We can represent the coefficients of each variable in each equation in matrix form, except the right-hand side, as:
[2 3]
[4 -5]
The determinant of the matrix is:
2*(-5) – 3*4 = -22
Now, we can replace the coefficients of x in the matrix with the right-hand side:
[8 3]
[-7 -5]
The determinant of this matrix is:
8*(-5) – 3*(-7) = -29
We can do the same for the coefficients of y:
[2 8]
[4 -7]
The determinant of this matrix is:
2*(-7) – 8*4 = -22
Finally, we can use Cramer’s rule to find the values of x and y:
x = -29/-22 = 29/22
y = -22/-22 = 1
Therefore, the solution to the system of equations is (x, y) = (29/22, 1).
The advantages of using Cramer’s rule are that it’s easy to remember and can be used for any number of variables. However, this method can be computationally expensive and time-consuming for larger systems.
Applications of system of equations and how to solve them
System of equations is widely used in real-world problems, such as:
- Calculating the optimal product mix to maximize profit for a company.
- Determining the ideal dosage of different drugs to minimize side effects in patients.
- Calculating the optimal strategy for a game to maximize winnings.
To solve these problems, we can use the different methods discussed in this article. For example, the graphical method can be used to find the equilibrium point for the optimal product mix, the elimination method can be used to find the ideal dosage for different drugs, and the matrix method can be used to find the optimal strategy for a game.
Conclusion
In this article, we discussed five different methods to solve system of equations, including the graphical, substitution, elimination, matrix, and Cramer’s rule. We also provided step-by-step examples to illustrate each method and explained their advantages and disadvantages. Finally, we showed how system of equations can be applied in real-world problems, and which method is best suited for each type of problem. We hope this article has helped you understand the concepts and master the skill of solving system of equations with ease.
Now it’s your turn to solve some systems of equations on your own.