Problem 56 - Introductory Algebra for College Students

Question

Explain how to determine if an ordered pair is a solution of a system of linear equations.

Step-by-Step Solution

Answer

An ordered pair is a solution of a system of linear equations if it satisfies all the equations in the system simultaneously. This can be checked by substitutng the x and y values from the ordered pair into each equation and ensuring the transformed equations hold true.

Step 1: Understanding the Concept of Ordered Pair and System of Equations

An ordered pair is a pair of two numbers in which the order of the numbers is significant, usually denoted as (x, y). A system of linear equations comprises two or more linear equations that share the same variables. If a particular ordered pair is a solution for the system, it means that when we substitute the values of x and y from the ordered pair into the system of equations, both equations (assuming a two-variable system) should be true simultaneously.

Step 2: Substitute the ordered pair into the first equation

The first step is to take the x and y coordinates from the ordered pair and substitute them into the first equation of the system. Replace the variable x with the first number in the ordered pair and y with the second number. Then, simplify the equation to see if both sides are equal.

Step 3: Substitute the ordered pair into the second equation

After checking the first equation, the next step is to substitute the ordered pair into the second equation of the system. Once again, substitute x and y with the values from the ordered pair and simplify to check if both sides of the equation are equal.

Step 4: Validate if the ordered pair is a solution of the system

If the ordered pair satisfactorily makes both equations true, then this ordered pair is indeed a solution of the system of linear equations. If not, then the given ordered pair fails to satisfy the entire system.