An ordered pair that satisfies all the equations in a system is known as a solution to the system. This solution represents a point where all the equations in the system intersect. In a two-variable system, the solution is typically represented as $(x, y)$, where $x$ and $y$ are the values that satisfy all the equations in the system.

Definition and Explanation

When we solve a system of equations, we are essentially finding the values of the variables that make all the equations in the system true simultaneously. An ordered pair that accomplishes this is referred to as a solution to the system. If the system has more than one variable, a solution will be an ordered pair of values, such as $(x, y)$.

Substitution Method

To verify if an ordered pair is a solution to the system of equations, follow these steps:

Substitute the $x$ value from the ordered pair into all the $x$ terms in the equations. Substitute the $y$ value from the ordered pair into all the $y$ terms in the equations. Check if the resulting statements are true. If all the statements are true, then the ordered pair is indeed a solution to the system.

For example, consider the system of equations:

$2x 3y 12$

$x - y 1$

If we have an ordered pair $(x, y) (3, 2)$, we can substitute these values into both equations:

$2(3) 3(2) 12 implies 6 6 12$ (True)

$3 - 2 1 implies 1 1$ (True)

Since both statements are true, $(3, 2)$ is a solution to the system.

When the System Has Multiple Solutions

If the system of equations has multiple solutions, each solution is represented by an ordered pair that satisfies all the equations. Therefore, each ordered pair in the solution set is a solution to the system. The solution set may contain more than one ordered pair, each fulfilling the same condition mentioned above.

For instance, consider the system:

$x y 5$

$2x 2y 10$

This system has infinitely many solutions, all of which lie on the line defined by $y 5 - x$. Some of these solutions include $(0, 5)$, $(1, 4)$, $(2, 3)$, etc. Each of these ordered pairs, when substituted into the system, will yield true statements.

Conclusion and Key Points

In summary, an ordered pair that satisfies all the equations in a system is called a solution. The ordered pairs in the solution set are those that satisfy the system when the values of the variables are substituted into the equations. The solution set can contain one or more ordered pairs, and each ordered pair is a solution to the system.

Key Terms:

Solution Set: The set of all ordered pairs that satisfy the system of equations. Ordered Pair: A pair of numbers that represent the values of the variables in the system of equations. System of Equations: A set of equations that share variables and need to be solved simultaneously.

To ensure that an ordered pair is a solution:

Substitute the values of the ordered pair into all the equations. Verify that each substituted equation is true.

By understanding the concept of a solution set and the process of verification, you can solve and verify systems of equations with confidence.