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Checking Solutions for Linear Systems of Equations
In mathematics, a system of linear equations is a collection of two or more linear equations that share the same variables. A solution to a system of linear equations is a set of values for the variables that satisfy all of the equations in the system.
For example, consider the following system of linear equations:
x + y = 5
2x - y = 1
The solution to this system is x = 2 and y = 3. This is because if we substitute these values into both equations, we get:
2 + 3 = 5
2(2) - 3 = 1
Both of these equations are true, so the solution x = 2 and y = 3 satisfies the system of equations.
How to Check Solutions for Linear Systems of Equations
To check if a given solution satisfies a system of linear equations, we can use the following steps:
- Substitute the values of the solution into each equation in the system.
- Simplify each equation.
- Check if the simplified equations are true.
If all of the simplified equations are true, then the solution satisfies the system of equations. If any of the simplified equations are false, then the solution does not satisfy the system of equations.
Example
Let's consider the following system of linear equations:
x + y + z = 6
2x - y + z = 3
x + 2y - z = 1
We want to check if the solution x = 1, y = 2, and z = 3 satisfies this system.
Substituting these values into the first equation, we get:
1 + 2 + 3 = 6
This equation is true, so the solution satisfies the first equation.
Substituting these values into the second equation, we get:
2(1) - 2 + 3 = 3
This equation is also true, so the solution satisfies the second equation.
Finally, substituting these values into the third equation, we get:
1 + 2(2) - 3 = 1
This equation is also true, so the solution satisfies the third equation.
Since the solution satisfies all three equations in the system, we can conclude that the solution x = 1, y = 2, and z = 3 is a valid solution to the system of equations.
Conclusion
Checking solutions for linear systems of equations is a simple process that involves substituting the values of the solution into each equation and verifying if the equations hold true. This method ensures that the solution satisfies all equations in the system.