How to Check a Solution to a System of Two Equations

In algebra, a system of equations represents a set of two or more equations that share the same variables. A solution to a system of equations is a set of values for the variables that satisfy all the equations simultaneously. This means that when you substitute these values into each equation, the equation becomes a true statement.

One way to check if a given point is a solution to a system of two equations is by substituting the coordinates of the point into both equations and checking if they satisfy both equations simultaneously. If they do, then the point is a solution to the system. If not, then the point is not a solution to the system.

Example:

Consider the following system of two equations:

Equation 1: x + 2y = 5

Equation 2: 3x – y = 1

Let’s check if the point (1, 2) is a solution to this system.

Substituting x = 1 and y = 2 into Equation 1, we get:

1 + 2(2) = 5

1 + 4 = 5

5 = 5

This is a true statement. Therefore, the point (1, 2) satisfies Equation 1.

Now, substituting x = 1 and y = 2 into Equation 2, we get:

3(1) – 2 = 1

3 – 2 = 1

1 = 1

This is also a true statement. Therefore, the point (1, 2) satisfies Equation 2.

Since the point (1, 2) satisfies both Equation 1 and Equation 2, it is a solution to the system of equations.

Steps to Check a Solution:

1. Substitute the x and y values of the given point into the first equation.
2. Simplify the equation and check if the equation holds true.
3. Repeat steps 1 and 2 for the second equation.
4. If both equations hold true after substituting the point, then the point is a solution to the system of equations.

Conclusion:

Checking if a point is a solution to a system of two equations is a simple process that involves substituting the coordinates of the point into both equations and verifying if they satisfy both equations simultaneously. This method is essential for understanding the concept of solutions to systems of equations and can be applied to any system of equations involving two or more variables.