Finding Ordered Pair Solutions For 3x + Y = 7
In mathematics, particularly in algebra, we often encounter equations with two variables, such as x and y. A solution to such an equation is an ordered pair (x, y) that, when substituted into the equation, makes the equation true. This article will delve into the process of finding ordered pair solutions for linear equations, using the example equation 3x + y = 7. We will explore different methods and strategies to systematically identify these solutions, providing a comprehensive understanding of this fundamental concept in algebra.
Understanding Linear Equations and Ordered Pairs
Before we dive into the methods for finding solutions, let's first define what we mean by a linear equation and an ordered pair. A linear equation in two variables is an equation that can be written in the form Ax + By = C, where A, B, and C are constants, and x and y are the variables. The graph of a linear equation is always a straight line.
An ordered pair is a pair of numbers, written in the form (x, y), where the order matters. The first number, x, represents the x-coordinate, and the second number, y, represents the y-coordinate. An ordered pair can be plotted as a point on a coordinate plane. For an ordered pair to be a solution to a linear equation, the values of x and y must satisfy the equation when substituted.
In the given equation, 3x + y = 7, we have a linear equation with A = 3, B = 1, and C = 7. Our goal is to find ordered pairs (x, y) that make this equation true.
Methods for Finding Ordered Pair Solutions
There are several methods we can use to find ordered pair solutions for linear equations. We'll explore two common approaches:
- Substitution Method: This method involves choosing a value for one variable (either x or y) and then substituting that value into the equation to solve for the other variable. This process yields an ordered pair (x, y) that satisfies the equation.
- Rearranging the Equation: This method involves rearranging the equation to isolate one variable in terms of the other. This allows us to easily generate solutions by choosing values for one variable and calculating the corresponding values for the other.
Let's apply these methods to the equation 3x + y = 7.
1. Substitution Method
With the substitution method, we pick a value for either x or y and substitute it into the equation to find the corresponding value of the other variable. For instance, let's start by choosing a value for x.
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Let x = 1: Substituting x = 1 into the equation 3x + y = 7, we get: 3(1) + y = 7 3 + y = 7 y = 7 - 3 y = 4 So, when x = 1, y = 4. This gives us the ordered pair (1, 4).
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Let x = 0: Substituting x = 0 into the equation, we get: 3(0) + y = 7 0 + y = 7 y = 7 So, when x = 0, y = 7. This gives us the ordered pair (0, 7).
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Let x = 2: Substituting x = 2 into the equation, we get: 3(2) + y = 7 6 + y = 7 y = 7 - 6 y = 1 So, when x = 2, y = 1. This gives us the ordered pair (2, 1).
We can continue this process with different values of x to find more ordered pair solutions. Alternatively, we can choose values for y and solve for x.
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Let y = 0: Substituting y = 0 into the equation, we get: 3x + 0 = 7 3x = 7 x = 7/3 So, when y = 0, x = 7/3. This gives us the ordered pair (7/3, 0).
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Let y = 1: Substituting y = 1 into the equation, we get: 3x + 1 = 7 3x = 7 - 1 3x = 6 x = 6/3 x = 2 So, when y = 1, x = 2. This gives us the ordered pair (2, 1), which we already found.
By using the substitution method, we can generate a variety of ordered pair solutions for the equation 3x + y = 7.
2. Rearranging the Equation
Another effective method is to rearrange the equation to isolate one variable in terms of the other. This makes it easier to find solutions by simply choosing values for one variable and calculating the corresponding value of the other.
In our equation, 3x + y = 7, we can isolate y by subtracting 3x from both sides:
y = 7 - 3x
Now, we have y expressed in terms of x. This form of the equation allows us to easily find solutions by choosing values for x and calculating the corresponding values for y.
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Let x = 0: Substituting x = 0 into the rearranged equation, we get: y = 7 - 3(0) y = 7 - 0 y = 7 So, when x = 0, y = 7. This gives us the ordered pair (0, 7).
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Let x = 1: Substituting x = 1 into the rearranged equation, we get: y = 7 - 3(1) y = 7 - 3 y = 4 So, when x = 1, y = 4. This gives us the ordered pair (1, 4).
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Let x = -1: Substituting x = -1 into the rearranged equation, we get: y = 7 - 3(-1) y = 7 + 3 y = 10 So, when x = -1, y = 10. This gives us the ordered pair (-1, 10).
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Let x = 7/3: Substituting x = 7/3 into the rearranged equation, we get: y = 7 - 3(7/3) y = 7 - 7 y = 0 So, when x = 7/3, y = 0. This gives us the ordered pair (7/3, 0).
As you can see, rearranging the equation provides a straightforward way to generate ordered pair solutions. We can choose any value for x, substitute it into the equation, and calculate the corresponding value for y.
Infinite Solutions
Linear equations in two variables have an infinite number of solutions. This is because for any value we choose for x, there is a corresponding value of y that satisfies the equation, and vice versa. The solutions can be represented graphically as points on a straight line. Each point on the line corresponds to an ordered pair solution of the equation.
Expressing the Solution Set
Since there are infinitely many solutions, we often express the solution set in a general form. Using the rearranged equation y = 7 - 3x, we can express the solution set as:
{ (x, 7 - 3x) | x is a real number }
This notation means that the solution set consists of all ordered pairs of the form (x, 7 - 3x), where x can be any real number. This concisely represents the infinite number of solutions to the equation.
Verification of Solutions
It's always a good practice to verify that the ordered pairs we find are indeed solutions to the equation. To do this, we substitute the values of x and y into the original equation and check if the equation holds true.
For example, let's verify the ordered pair (1, 4) for the equation 3x + y = 7:
3(1) + 4 = 7 3 + 4 = 7 7 = 7
The equation holds true, so (1, 4) is indeed a solution.
Let's verify the ordered pair (0, 7):
3(0) + 7 = 7 0 + 7 = 7 7 = 7
The equation holds true, so (0, 7) is also a solution.
Let's verify the ordered pair (7/3, 0):
3(7/3) + 0 = 7 7 + 0 = 7 7 = 7
The equation holds true, so (7/3, 0) is a solution as well.
By verifying our solutions, we can ensure that we have correctly identified ordered pairs that satisfy the equation.
Practical Applications
Finding ordered pair solutions for linear equations is a fundamental skill in algebra and has numerous practical applications in various fields. Linear equations are used to model relationships between two variables in many real-world scenarios, such as:
- Economics: Supply and demand curves, cost and revenue functions.
- Physics: Motion with constant velocity, linear relationships between force and displacement.
- Engineering: Circuit analysis, structural design.
- Computer Science: Linear programming, data analysis.
Understanding how to find solutions to linear equations allows us to analyze and solve problems in these fields. For instance, we can use linear equations to determine the break-even point for a business, predict the trajectory of a projectile, or optimize resource allocation in a project.
Conclusion
Finding ordered pair solutions for linear equations is a core concept in algebra. We've explored two primary methods – the substitution method and rearranging the equation – for systematically identifying these solutions. Both methods allow us to generate ordered pairs (x, y) that, when substituted into the equation, make it true.
Linear equations in two variables have an infinite number of solutions, which can be represented graphically as points on a straight line. We can express the solution set in a general form using set notation. It's crucial to verify solutions to ensure accuracy.
The ability to find ordered pair solutions for linear equations is a fundamental skill with wide-ranging applications in various disciplines, making it an essential concept to master in mathematics.
In summary, to find an ordered pair (x, y) that is a solution to the equation 3x + y = 7, we can use either the substitution method or rearrange the equation. By choosing a value for one variable and solving for the other, we can generate an ordered pair that satisfies the equation. For instance, if we let x = 1, we find y = 4, giving us the solution (1, 4). This process can be repeated to find infinitely many solutions, as linear equations in two variables have an infinite number of ordered pair solutions.
