A system of equations refers to a set of two or more equations that are related to one another or have same variables and must be solved together in order to find the values of the unknown variables that satisfy all the equations simultaneously.
These equations are often used to model real-world situations, and solving the system of equations can provide valuable insights or solutions to the problem at hand. The solution to a system of equations is typically a set of values that make all the equations in the system true at the same time.
How To Solve A System Of Equations
To solve a system of equations, we want to find the values of the variables that make all the equations true at the same time.
There are different methods to solve a system of equations, such as substitution and elimination. In this summary, we will explain how these methods work and when to use them.
Substitution method:
The substitution method involves solving one of the equations for one variable, and then plugging that expression into the other equation. This way, we eliminate one variable and get an equation with only one variable left.
To use the substitution method, we need to find an equation that is easy to solve for one variable, such as one that has a coefficient of 1 or -1.
For example, if we have the system:
x + y = 5
2x – y = 1
We can solve the first equation for y by subtracting x from both sides:
y = 5 – x
Then, we can substitute this expression for y into the second equation and simplify:
2x – (5 – x) = 1
3x – 5 = 1
3x = 6
x = 2
Once we have the value of one variable, we can plug it back into either equation and solve for the other variable. For example, using the first equation:
x + y = 5
2 + y = 5
y = 3
The solution of the system is the ordered pair (x, y) = (2, 3), which means that x = 2 and y = 3 make both equations true.
Elimination method:
The elimination method involves adding or subtracting the equations to eliminate one variable and get an equation with only one variable left.
To use the elimination method, we need to find a way to make the coefficients of one variable equal or opposite in both equations. For example, if we have the system:
3x + 2y = 8
4x – 2y = 10
We can see that the coefficients of y are opposite, so we can add the equations and eliminate y:
(3x + 2y) + (4x – 2y) = 8 + 10
7x = 18
x = 18/7
Once we have the value of one variable, we can plug it back into either equation and solve for the other variable.
For example, using the first equation:
3x + 2y = 8
3(18/7) + 2y = 8
54/7 + 2y = 8
2y = 8 – 54/7
2y = 2/7
y = 1/7
The solution of the system is the ordered pair (x, y) = (18/7, 1/7), which means that x = 18/7 and y = 1/7 make both equations true.
When to use each method
Both methods can be used to solve any system of equations, but some systems may be easier to solve with one method than the other. Here are some tips to help you choose the best method for a given system:
- If one of the equations is already solved for one variable, or can be easily solved for one variable, use the substitution method.
- If the coefficients of one variable are equal or opposite in both equations, use the elimination method.
- If the coefficients of both variables are fractions, try to clear the fractions by multiplying both equations by a common denominator, and then use either method.
- If the system has more than two equations or more than two variables, try to eliminate one variable at a time until you get a system with two equations and two variables, and then use either method.