Is The Ordered Pair A Solution? System Of Equations Explained
Have you ever wondered how to check if a specific pair of numbers actually solves a set of equations? This is a fundamental concept in algebra, and it's super useful for understanding how systems of equations work. In this guide, we'll break down the process step by step, using the example you provided. So, let's dive in and learn how to determine if an ordered pair is a solution to a system of equations!
Understanding Systems of Equations and Ordered Pairs
Before we get into the nitty-gritty, let's make sure we're on the same page about what systems of equations and ordered pairs are. A system of equations is simply a set of two or more equations that we're considering together. Think of it as a puzzle where we need to find values that satisfy all the equations at the same time. In our case, we have the following system:
9x + 3y = 27
-6x + 3y = -3
An ordered pair is a pair of numbers, usually written in the form (x, y), that represents a point on a coordinate plane. The first number, x, tells us how far to move horizontally, and the second number, y, tells us how far to move vertically. For example, the ordered pair (2, 3) means we move 2 units to the right and 3 units up.
The big question is: how do we know if an ordered pair is a solution to a system of equations? The answer is surprisingly simple: we plug the values into the equations and see if they hold true. If the ordered pair satisfies all equations in the system, then it's a solution. If it fails even one equation, it's not a solution. This process is straightforward but crucial for grasping the essence of solving systems of equations.
Remember, a solution to a system of equations is an ordered pair that makes all equations true simultaneously. This concept is the cornerstone of solving various mathematical problems and real-world applications. From determining the intersection point of two lines to solving complex problems in physics and economics, understanding how to verify solutions is indispensable. So, let's roll up our sleeves and see how this works with our specific examples!
Step-by-Step: Checking Ordered Pair (2, 3)
Let's start with the first ordered pair, (2, 3). This means x = 2 and y = 3. To check if this is a solution to our system of equations, we'll substitute these values into each equation and see if they balance.
Equation 1: 9x + 3y = 27
Replace x with 2 and y with 3:
9(2) + 3(3) = 27
Now, let's simplify:
18 + 9 = 27
And we get:
27 = 27
Great! The first equation holds true. But remember, the ordered pair needs to satisfy both equations to be a solution. So, let's move on to the second equation.
Equation 2: -6x + 3y = -3
Again, substitute x = 2 and y = 3:
-6(2) + 3(3) = -3
Simplify:
-12 + 9 = -3
And we have:
-3 = -3
Excellent! The second equation also holds true. Since the ordered pair (2, 3) satisfies both equations in the system, we can confidently say that it is a solution.
This meticulous step-by-step approach ensures accuracy and builds a solid understanding of the solution verification process. By substituting the values and simplifying, we clearly demonstrate whether the ordered pair fits the system of equations. This method is not just about getting the right answer; it's about understanding the underlying principles of algebraic solutions.
Step-by-Step: Checking Ordered Pair (7, 2/3)
Now, let's tackle the second ordered pair, (7, 2/3). This means x = 7 and y = 2/3. We'll follow the same process as before, substituting these values into each equation and checking for balance.
Equation 1: 9x + 3y = 27
Substitute x = 7 and y = 2/3:
9(7) + 3(2/3) = 27
Simplify:
63 + 2 = 27
This gives us:
65 = 27
Oops! This is not true. 65 does not equal 27. So, the first equation is not satisfied. We don't even need to check the second equation. If an ordered pair fails even one equation in the system, it's not a solution.
Therefore, the ordered pair (7, 2/3) is not a solution to the system of equations.
This example highlights the importance of verifying the solution against all equations in the system. A solution must satisfy each equation to be valid. By identifying that (7, 2/3) fails in the first equation itself, we save time and effort, reinforcing the core concept of solution verification.
Why This Matters: Real-World Applications
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