Is Roxanne's System Of Equations Solution Correct?

Hey guys, let's dive into the world of graphing systems of equations today! Roxanne tackled a problem where she had to find the solution to the following system:

$y=\frac{2}{3} x-5$ $y=-2 x+3$

She landed on the solution $(-3, -3)$ and is asking if she's spot on. Let's break down how to check this and figure out if Roxanne nailed it, or if there's a little mix-up happening. Finding the solution to a system of equations means finding the point $(x, y)$ where both lines intersect on the graph. This single point satisfies both equations simultaneously. When you graph two linear equations, they can either intersect at one point (a unique solution), be parallel and never intersect (no solution), or be the exact same line (infinitely many solutions). Roxanne's goal was to find that sweet spot where these two specific lines cross paths. To do this, she likely graphed both equations, carefully plotting points or using the slope-intercept form ($y = mx + b$, where $m$ is the slope and $b$ is the y-intercept) to draw each line. The point where the lines visually cross is her proposed solution. But how do we confirm if that point is truly the solution? There are two main ways: plugging the coordinates back into the original equations or analyzing the graphical representation. We'll focus on the algebraic check because it's the most definitive way to confirm accuracy, regardless of how perfectly the graph was drawn. So, Roxanne found $(-3, -3)$ and believes it's the answer. Let's put her answer to the test using the equations she was given. This process is super important for understanding how solutions work and for double-checking your own math homework. Don't worry if you've made similar mistakes before; we've all been there! The key is to learn from them and get better. So, grab your pencils (or keyboards!) and let's get to the bottom of Roxanne's solution.

The Importance of Verification in Solving Systems

Alright, so Roxanne thinks her solution to the system $y=\frac{2}{3} x-5$ and $y=-2 x+3$ is $(-3, -3)$. Now, the million-dollar question: is she right? The most crucial step after finding a potential solution, especially when graphing, is verification. Graphing can be a bit tricky sometimes. Slight inaccuracies in drawing the lines, misinterpreting the scale, or even just a shaky hand can lead you to a point that looks like the intersection but isn't exactly it. That's why relying solely on the visual can be a trap. To be absolutely sure, we need to substitute the proposed solution's $x$ and $y$ values into both original equations. If the point $(-3, -3)$ makes both equations true statements, then Roxanne is correct. If it only makes one true, or neither, then it's not the solution. This verification process is a cornerstone of algebraic problem-solving. It's not just about getting the answer; it's about proving you have the right answer. Think of it like a detective checking all the evidence. You can't just assume the first clue is the right one; you have to cross-reference and confirm. Let's apply this detective work to Roxanne's proposed solution $(-3, -3)$. We'll take the $x$-value, which is $-3$, and the $y$-value, which is also $-3$, and plug them into each equation. This is where we'll find out if Roxanne's hard work paid off or if there's a detail she might have missed. This is also where we can address common errors, like accidentally switching the $x$ and $y$ coordinates when plugging them in, which is a frequent pitfall for many students. So, let's meticulously go through each equation to see if $(-3, -3)$ holds true.

Testing the First Equation: $y=\frac{2}{3} x-5$

First up, let's take Roxanne's proposed solution, $(-3, -3)$, and plug it into the first equation: $y=\frac{2}{3} x-5$. Remember, the $x$-coordinate is $-3$ and the $y$-coordinate is $-3$. So, we replace $y$ with $-3$ and $x$ with $-3$.

$-3 = \frac{2}{3} (-3) - 5$

Now, let's simplify the right side of the equation. We multiply $\frac{2}{3}$ by $-3$. The $3$ in the denominator cancels out with the $-3$, leaving us with $2 \times -1$, which equals $-2$.

$-3 = -2 - 5$

Next, we combine the numbers on the right side: $-2 - 5$ equals $-7$.

$-3 = -7$

Looking at this statement, $-3 = -7$, we can see that this is false. The left side does not equal the right side. This immediately tells us that the point $(-3, -3)$ does not satisfy the first equation. If a point doesn't even work in one of the equations, it cannot be the solution to the system. The solution must satisfy all equations in the system. So, based on this test alone, Roxanne's solution is incorrect. However, to be thorough and to understand why it might be incorrect or if there's another issue, we should also check the second equation. Sometimes, a mistake might lead to a point that works for one equation but not the other, and understanding which one fails can give clues about the nature of the error.

Testing the Second Equation: $y=-2 x+3$

Let's continue our verification process by plugging Roxanne's proposed solution, $(-3, -3)$, into the second equation: $y=-2 x+3$. Again, we substitute $y = -3$ and $x = -3$.

$-3 = -2 (-3) + 3$

Now, we simplify the right side. Multiplying $-2$ by $-3$ gives us $6$ (a negative times a negative is a positive).

$-3 = 6 + 3$

Next, we add $6$ and $3$ on the right side:

$-3 = 9$

This statement, $-3 = 9$, is also false. The point $(-3, -3)$ does not satisfy the second equation either. This confirms our earlier finding from the first equation: Roxanne's proposed solution is definitely incorrect. It fails to make either equation true. This means the point $(-3, -3)$ does not lie on either of the lines represented by these equations. Therefore, it cannot be the point where the lines intersect.

Why Roxanne Might Be Incorrect

So, we've established that Roxanne's solution $(-3, -3)$ is incorrect because it doesn't satisfy either equation. Now, let's think about why this might have happened. The most common reasons for errors when solving systems of equations graphically are:

1. Inaccurate Graphing: As mentioned, drawing lines perfectly can be challenging. If the grid isn't clear, or if the slope is a fraction like $\frac{2}{3}$, it's easy to misplace the line. This could lead Roxanne to read the intersection point incorrectly from her graph.
2. Calculation Errors: Even if the graph was perfect, when Roxanne went to calculate the intersection point algebraically (or verify her graphical solution), she might have made a mistake in her arithmetic. The calculations we did above show that $(-3, -3)$ doesn't work, suggesting an error occurred somewhere.
3. Switching x and y Values: This is a very common mistake, and it's actually hinted at in option B of the question. Sometimes, students correctly solve for $x$ and $y$ but then write the coordinate pair in the wrong order, like $(y, x)$ instead of $(x, y)$. Or, when plugging values back into equations for verification, they might swap them. Let's explore this possibility.

Did Roxanne Switch the x and y Values?

Let's consider the possibility that Roxanne did find the correct values for $x$ and $y$ but reported them in the wrong order. What if the actual solution had an $x$-coordinate of $-3$ and a $y$-coordinate of $-3$, but the values were mixed up in some way that led to $(-3,-3)$ being reported? This is a bit circular, as $(-3, -3)$ has the same $x$ and $y$ value, so switching them doesn't change the point. However, the reason she might have gotten $(-3,-3)$ could be due to a switch during calculation or plotting. For example, if the actual solution was $(a, b)$, she might have ended up calculating or plotting $(b, a)$ and then reporting that.

Let's find the actual solution to the system algebraically. We can use substitution or elimination. Since both equations are already solved for $y$, substitution is straightforward.

Set the expressions for $y$ equal to each other:

$\frac{2}{3} x - 5 = -2x + 3$

To get rid of the fraction, multiply the entire equation by 3:

$3 \times (\frac{2}{3} x - 5) = 3 \times (-2x + 3)$

$2x - 15 = -6x + 9$

Now, let's gather the $x$ terms on one side and the constants on the other. Add $6x$ to both sides:

$2x + 6x - 15 = 9$

$8x - 15 = 9$

Add $15$ to both sides:

$8x = 9 + 15$

$8x = 24$

Divide by 8:

$x = \frac{24}{8}$

$x = 3$

So, the $x$-coordinate of the actual solution is $3$. Now, let's find the $y$-coordinate by plugging $x=3$ into either of the original equations. Let's use the second one, $y = -2x + 3$, as it's simpler:

$y = -2(3) + 3$

$y = -6 + 3$

$y = -3$

So, the actual solution to the system of equations is $(3, -3)$.

Now we can see what happened. Roxanne's proposed solution was $(-3, -3)$, but the correct solution is $(3, -3)$. The $y$-values match, but the $x$-values are different ($3$ vs. $-3$). It's very likely that Roxanne made a sign error when solving for $x$ or when plotting the point. She might have correctly found that $x=3$ but written it as $-3$, or perhaps when graphing, she plotted $x=-3$ on the x-axis instead of $x=3$. The fact that the $y$-value is correct is interesting. Let's re-check the first equation with $x=3$ and $y=-3$: $y = \frac{2}{3}x - 5 ightarrow -3 = \frac{2}{3}(3) - 5 ightarrow -3 = 2 - 5 ightarrow -3 = -3$. It works! And the second equation: $y = -2x + 3 ightarrow -3 = -2(3) + 3 ightarrow -3 = -6 + 3 ightarrow -3 = -3$. It also works! So, the point $(3, -3)$ definitely solves the system.

Conclusion: Was Roxanne Correct?

Based on our detailed verification, Roxanne's solution of $(-3, -3)$ is incorrect. We plugged her proposed solution into both original equations:

• For $y=\frac{2}{3} x-5$: $-3 = \frac{2}{3}(-3) - 5$ simplifies to $-3 = -7$, which is false.
• For $y=-2 x+3$: $-3 = -2(-3) + 3$ simplifies to $-3 = 9$, which is false.

Since the point $(-3, -3)$ does not satisfy either equation, it cannot be the point of intersection. We also went ahead and found the actual solution to be $(3, -3)$. It appears Roxanne likely made a sign error when determining the $x$-coordinate, possibly confusing $3$ with $-3$. The option provided that best describes this situation is B. No. She switched the $x$ and $y$ values of the solution. While it wasn't strictly switching $x$ and $y$ (as they were both $-3$ in her answer), the error was likely a sign error related to the $x$-value, which is a common type of mix-up, potentially leading to a perceived switch or incorrect value. If the correct solution was, for example, $(3, -7)$ and she reported $(-7, 3)$, then option B would be a direct explanation. In this specific case, her $x$-value was wrong (it should be $3$, not $-3$). A sign error is a form of misrepresenting the correct value, similar to how switching $x$ and $y$ misrepresents the correct coordinate pair. Therefore, option B, which points to a mix-up of coordinate values, is the most fitting explanation among the choices, given the commonality of such errors in graphing and solving systems.