Determine Whether (8,2) Is A Solution Of 3x - 3y = 1

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In the realm of mathematics, a fundamental concept involves determining whether a given point satisfies a specific equation. This often arises when working with linear equations and their graphical representations. In this comprehensive article, we will delve into the process of verifying whether the point $(8,2)$ is a solution to the linear equation $3x - 3y = 1$. Understanding this process is crucial for grasping the relationship between algebraic equations and their geometric counterparts, which is a cornerstone of algebra and coordinate geometry. Before we jump into the specifics, let’s establish the core principles that govern the solutions of linear equations and how points in the coordinate plane relate to these equations. We will explore the fundamental concept of ordered pairs, substitution, and how these concepts interplay to determine if a given point lies on the line represented by the equation. This exploration will not only answer our initial question but also provide a solid foundation for tackling similar problems in algebra and beyond.

Understanding Solutions to Linear Equations

When we talk about a solution to a linear equation, we're referring to a point that, when its coordinates are substituted into the equation, makes the equation true. A linear equation, in its simplest form, can be represented as $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. Graphically, a linear equation represents a straight line on a coordinate plane. Each point on this line corresponds to an ordered pair $(x, y)$ that satisfies the equation. Conversely, any ordered pair that satisfies the equation represents a point that lies on the line. This is a fundamental concept in coordinate geometry, linking algebra and geometry in a powerful way. To determine if a point is a solution, we substitute the $x$ and $y$ values of the point into the equation. If the left-hand side of the equation equals the right-hand side after the substitution, then the point is indeed a solution. This process is straightforward but requires careful attention to arithmetic and algebraic manipulation. The beauty of this method lies in its simplicity and universality; it applies to any linear equation and any point in the coordinate plane. By mastering this technique, one gains a valuable tool for solving a wide range of problems in mathematics and related fields.

The Point $(8,2)$ and the Equation $3x - 3y = 1$

Now, let's focus on the specific question at hand: Is the point $(8,2)$ a solution to the equation $3x - 3y = 1$? To answer this, we will employ the substitution method, which is the most direct and reliable way to verify the solution. The point $(8,2)$ is an ordered pair where $x = 8$ and $y = 2$. We will substitute these values into the equation $3x - 3y = 1$ and simplify. The equation becomes $3(8) - 3(2) = 1$. Performing the multiplication, we get $24 - 6 = 1$. Simplifying further, we have $18 = 1$. This statement is clearly false. The left-hand side, 18, does not equal the right-hand side, 1. Therefore, the point $(8,2)$ does not satisfy the equation $3x - 3y = 1$. This means that the point $(8,2)$ does not lie on the line represented by the equation $3x - 3y = 1$. This exercise highlights the importance of accurate substitution and simplification in determining the validity of a solution. It also reinforces the connection between algebraic equations and their graphical representations. Understanding this connection is crucial for developing a deeper understanding of mathematics.

Step-by-Step Substitution and Verification

To further clarify the process, let's break down the substitution and verification steps in detail. This step-by-step approach is crucial for avoiding errors and ensuring accuracy when dealing with mathematical problems.

  1. Identify the equation and the point: Our equation is $3x - 3y = 1$, and our point is $(8,2)$.
  2. Substitute the x-coordinate: Replace $x$ in the equation with the x-coordinate of the point, which is 8. This gives us $3(8) - 3y = 1$.
  3. Substitute the y-coordinate: Replace $y$ in the equation with the y-coordinate of the point, which is 2. Now we have $3(8) - 3(2) = 1$.
  4. Perform the multiplication: Multiply the constants with the numbers inside the parentheses: $24 - 6 = 1$.
  5. Simplify the equation: Subtract 6 from 24: $18 = 1$.
  6. Compare the two sides: The left-hand side (18) does not equal the right-hand side (1). Therefore, the equation is not satisfied.
  7. Conclusion: The point $(8,2)$ is not a solution to the equation $3x - 3y = 1$.

This methodical approach underscores the importance of meticulousness in mathematical calculations. Each step is crucial, and any error along the way can lead to an incorrect conclusion. By following this step-by-step process, we can confidently determine whether a given point is a solution to a linear equation.

Visualizing the Equation and the Point

To enhance our understanding, let's visualize the equation $3x - 3y = 1$ and the point $(8,2)$ on a coordinate plane. Visualizing mathematical concepts can often provide a more intuitive grasp of the underlying principles. The equation $3x - 3y = 1$ represents a straight line. To plot this line, we need to find at least two points that satisfy the equation. We can do this by choosing arbitrary values for $x$ and solving for $y$, or vice versa. For example, if we let $x = 0$, the equation becomes $-3y = 1$, which gives us $y = -1/3$. So, the point $(0, -1/3)$ is on the line. If we let $y = 0$, the equation becomes $3x = 1$, which gives us $x = 1/3$. So, the point $(1/3, 0)$ is also on the line. By plotting these two points and drawing a line through them, we can visualize the graph of the equation $3x - 3y = 1$. Now, let's plot the point $(8,2)$ on the same coordinate plane. We can see that the point $(8,2)$ does not lie on the line represented by the equation $3x - 3y = 1$. This visual confirmation reinforces our earlier algebraic conclusion that $(8,2)$ is not a solution to the equation. The graphical representation provides a clear and immediate understanding of why the point does not satisfy the equation, further solidifying the connection between algebra and geometry.

Alternative Methods for Verification

While substitution is the most direct method for verifying a solution, it's worth noting that other approaches can also be used. Exploring alternative methods can deepen our understanding of the underlying mathematical principles and provide additional problem-solving tools. One such method involves rearranging the equation into slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Rearranging $3x - 3y = 1$, we first subtract $3x$ from both sides to get $-3y = -3x + 1$. Then, we divide both sides by -3 to get $y = x - 1/3$. Now, we can substitute the x-coordinate of the point $(8,2)$ into this equation to find the corresponding y-value on the line. Substituting $x = 8$, we get $y = 8 - 1/3 = 23/3$. Since the y-coordinate of the given point is 2, which is not equal to 23/3, we can conclude that the point $(8,2)$ does not lie on the line. Another approach involves calculating the distance between the point and the line. If the distance is greater than zero, then the point is not on the line. However, this method is more complex and typically used when a precise distance is required rather than a simple verification. By understanding these alternative methods, we can appreciate the flexibility and interconnectedness of mathematical concepts. Each method offers a different perspective on the problem and can be valuable in different contexts.

Conclusion: $(8,2)$ is Not a Solution

In conclusion, through our detailed exploration and step-by-step verification process, we have definitively determined that the point $(8,2)$ is not a solution to the equation $3x - 3y = 1$. We arrived at this conclusion through algebraic substitution, where we replaced $x$ and $y$ in the equation with the coordinates of the point and found that the resulting statement was false. We also reinforced this conclusion with a visual representation, plotting the line and the point on a coordinate plane and observing that the point did not lie on the line. Furthermore, we discussed alternative methods for verification, highlighting the versatility of mathematical approaches. This exercise underscores the fundamental concept of what it means for a point to be a solution to a linear equation and the importance of accurate substitution and simplification in mathematical problem-solving. The ability to verify solutions is a crucial skill in algebra and beyond, providing a foundation for more advanced mathematical concepts. By mastering this skill, we gain confidence in our ability to solve equations and understand their graphical representations. The exploration of this specific problem serves as a valuable case study, illustrating the general principles and techniques applicable to a wide range of mathematical problems.