How to Check a Solution to a System of Two Equations

How to Check a Solution to a System of Two Equations

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously. A solution to a system of equations is a set of values for the variables that satisfy all the equations in the system. In this article, we will focus on how to check if a given point is a solution to a system of two linear equations.

What is a System of Two Equations?

A system of two linear equations typically looks like this:

Equation 1: ax + by = c

Equation 2: dx + ey = f

Where a, b, c, d, e, and f are constants, and x and y are variables. A solution to this system is a pair of values (x, y) that satisfies both Equation 1 and Equation 2.

Checking a Solution

To check if a given point (x₁, y₁) is a solution to a system of two equations, follow these steps:

1. Substitute the values of x₁ and y₁ into Equation 1.
2. Simplify the equation and check if the left-hand side equals the right-hand side.
3. Repeat steps 1 and 2 for Equation 2.

If the point (x₁, y₁) satisfies both equations, then it is a solution to the system. If it does not satisfy at least one of the equations, then it is not a solution.

Example

Let's consider the following system of equations:

Equation 1: 2x + 3y = 7

Equation 2: x - y = 1

We want to check if the point (2, 1) is a solution to this system.

Step 1: Substitute the values of x₁ = 2 and y₁ = 1 into Equation 1.

2(2) + 3(1) = 7

Step 2: Simplify the equation.

4 + 3 = 7

7 = 7

The left-hand side equals the right-hand side. Therefore, the point (2, 1) satisfies Equation 1.

Step 3: Repeat steps 1 and 2 for Equation 2.

2 - 1 = 1

1 = 1

The left-hand side equals the right-hand side. Therefore, the point (2, 1) satisfies Equation 2.

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

Conclusion

Checking if a given point is a solution to a system of two equations is a straightforward process. By substituting the coordinates of the point into each equation and verifying if they satisfy both equations simultaneously, we can determine if the point is a solution. This process is crucial for understanding the concept of solutions in systems of equations and is applicable to various mathematical problems.