Which Of The Following Ordered Pairs Is A Solution To The Linear Equation X - 2y = 4? A) (10, -1) B) (10, 2) C) (6, 1) D) (6, 2) E) (10, 1).
In mathematics, particularly in algebra, linear equations play a fundamental role. These equations, which represent straight lines when graphed, are essential tools for modeling real-world scenarios and solving various problems. One crucial aspect of working with linear equations is identifying their solutions. A solution to a linear equation in two variables (typically denoted as x and y) is an ordered pair (x, y) that, when substituted into the equation, makes the equation true. This article delves into the process of determining whether a given ordered pair is a solution to a specific linear equation. We will use the linear equation x - 2y = 4 as our example and explore several ordered pairs to ascertain which ones satisfy this equation. Understanding how to find solutions to linear equations is a cornerstone of algebraic proficiency, paving the way for more advanced mathematical concepts and applications.
Before we dive into specific ordered pairs, let's solidify our understanding of linear equations and their solutions. A linear equation in two variables can be written in the general 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. A solution to a linear equation is an ordered pair (x, y) that satisfies the equation. This means that when we substitute the x-coordinate and the y-coordinate of the ordered pair into the equation, the left-hand side of the equation equals the right-hand side. In other words, the equation holds true. For example, in the equation x - 2y = 4, we are looking for pairs of numbers that, when x is substituted and y is doubled and subtracted from x, result in 4. The process of verifying whether an ordered pair is a solution involves substituting the given values of x and y into the equation and checking if the equality holds. This is a fundamental skill in algebra and is crucial for solving systems of equations, graphing lines, and understanding the relationships between variables. By mastering this concept, students can build a strong foundation for more advanced mathematical topics and real-world applications.
Now, let's apply our understanding to the specific equation x - 2y = 4. We are given a set of ordered pairs, and our task is to determine which of these pairs are solutions to the equation. The given options are:
a) (10, -1) b) (10, 2) c) (6, 1) d) (6, 2) e) (10, 1)
To find the solutions, we will substitute the x and y values from each ordered pair into the equation and check if the equation holds true. This process involves simple arithmetic operations: substitution, multiplication, and subtraction. We will meticulously evaluate each ordered pair to ensure accuracy. The goal is to identify the pairs that, when their values are plugged into the equation, result in the left side being equal to the right side (which is 4 in this case). This method is a direct application of the definition of a solution to a linear equation. It is a systematic approach that guarantees we find all the correct solutions from the given options. This exercise not only helps in understanding the concept of solutions but also reinforces the importance of careful calculation and attention to detail in mathematical problem-solving.
Let's systematically verify each ordered pair:
a) (10, -1)
Substitute x = 10 and y = -1 into the equation x - 2y = 4:
10 - 2(-1) = 10 + 2 = 12
Since 12 ≠ 4, the ordered pair (10, -1) is not a solution.
b) (10, 2)
Substitute x = 10 and y = 2 into the equation x - 2y = 4:
10 - 2(2) = 10 - 4 = 6
Since 6 ≠ 4, the ordered pair (10, 2) is not a solution.
c) (6, 1)
Substitute x = 6 and y = 1 into the equation x - 2y = 4:
6 - 2(1) = 6 - 2 = 4
Since 4 = 4, the ordered pair (6, 1) is a solution.
d) (6, 2)
Substitute x = 6 and y = 2 into the equation x - 2y = 4:
6 - 2(2) = 6 - 4 = 2
Since 2 ≠ 4, the ordered pair (6, 2) is not a solution.
e) (10, 1)
Substitute x = 10 and y = 1 into the equation x - 2y = 4:
10 - 2(1) = 10 - 2 = 8
Since 8 ≠ 4, the ordered pair (10, 1) is not a solution.
To further clarify why some ordered pairs are solutions while others are not, let's delve into a more detailed analysis of each option. Understanding the specific calculations and how they relate to the equation x - 2y = 4 is crucial for mastering this concept.
a) (10, -1): When we substitute x = 10 and y = -1, the equation becomes 10 - 2(-1). The multiplication of -2 and -1 results in +2, so the expression simplifies to 10 + 2, which equals 12. Since 12 is not equal to 4, this ordered pair does not satisfy the equation. The key here is to pay attention to the negative signs and the order of operations. This example highlights the importance of accurate arithmetic in determining solutions.
b) (10, 2): Substituting x = 10 and y = 2 into the equation gives us 10 - 2(2). Multiplying 2 by 2 yields 4, so the expression becomes 10 - 4, which equals 6. Again, 6 is not equal to 4, indicating that (10, 2) is not a solution. This example reinforces the need for precise calculations to avoid errors in the evaluation process.
c) (6, 1): This is the solution we identified earlier. When we substitute x = 6 and y = 1, the equation becomes 6 - 2(1). Multiplying 2 by 1 gives 2, and then subtracting 2 from 6 results in 4. Since 4 is equal to 4, this ordered pair satisfies the equation. This example demonstrates the direct application of the solution definition, where the substituted values make the equation true.
d) (6, 2): Substituting x = 6 and y = 2 into the equation yields 6 - 2(2). Multiplying 2 by 2 gives 4, so the expression becomes 6 - 4, which equals 2. As 2 is not equal to 4, the ordered pair (6, 2) is not a solution. This example further emphasizes the importance of accurate arithmetic and careful substitution.
e) (10, 1): When we substitute x = 10 and y = 1, the equation becomes 10 - 2(1). Multiplying 2 by 1 gives 2, and subtracting 2 from 10 results in 8. Since 8 is not equal to 4, the ordered pair (10, 1) is not a solution. This example serves as another reminder of the need for meticulous calculations and a clear understanding of the equation's structure.
By analyzing each option in detail, we gain a deeper understanding of how ordered pairs interact with linear equations and what it truly means for a pair to be a solution. This step-by-step approach not only helps in solving the specific problem but also builds a stronger foundation for future algebraic tasks.
Through the process of substituting and evaluating, we have determined that the ordered pair (6, 1) is the only solution among the given options for the linear equation x - 2y = 4. This exercise demonstrates the fundamental method for verifying solutions to linear equations. By substituting the x and y values of an ordered pair into the equation and checking if the equation holds true, we can definitively determine whether the pair is a solution. This skill is crucial for various algebraic tasks, including solving systems of equations, graphing lines, and modeling real-world problems. Mastering the concept of solutions to linear equations is a cornerstone of algebraic proficiency, paving the way for more advanced mathematical concepts and applications.
The final answer is (c) (6, 1)