Solving A System Of Equations And Ratios
Hey guys! Today, we're diving into a fun math problem that involves solving a system of equations combined with ratios. It might sound intimidating, but trust me, we'll break it down step by step so it's super easy to understand. Our main goal is to find the solution to the system of equations where 2x - y = -3 and the ratio x/y = 6/8. We have a few options to choose from, and we need to figure out which one is the real deal.
Understanding the Problem
Before we jump into solving, let's make sure we understand exactly what we're dealing with. We have two equations:
- 2x - y = -3
- x/y = 6/8
The first equation is a linear equation. It tells us that if we multiply x by 2 and then subtract y, we should get -3. The second equation gives us a ratio between x and y. It says that x is to y as 6 is to 8. This can also be simplified to x/y = 3/4. Understanding these relationships is key to finding the right values for x and y that satisfy both equations.
Simplifying the Ratio
First, let's simplify the ratio x/y = 6/8. Both 6 and 8 are divisible by 2, so we can reduce the fraction:
6/8 = (6 Ă· 2) / (8 Ă· 2) = 3/4
So, our simplified ratio is x/y = 3/4. This means that x = (3/4)y. This form is super useful because it expresses x in terms of y, which we can then substitute into our first equation.
Substituting into the Linear Equation
Now that we have x = (3/4)y, we can substitute this into the first equation, 2x - y = -3. This will allow us to solve for y:
2((3/4)y) - y = -3
Multiply out the terms:
(6/4)y - y = -3
Simplify the fraction:
(3/2)y - y = -3
To combine the y terms, we need a common denominator. We can rewrite y as (2/2)y:
(3/2)y - (2/2)y = -3
Now, subtract the y terms:
(1/2)y = -3
To solve for y, multiply both sides by 2:
y = -3 * 2 = -6
Solving for x
Now that we have the value of y, we can find the value of x using the simplified ratio x = (3/4)y:
x = (3/4) * (-6)
Multiply the terms:
x = -18/4
Simplify the fraction:
x = -9/2 = -4.5
So, we have x = -4.5 and y = -6. However, looking at our options, we see that none of them match these values. This suggests we might need to re-evaluate our steps or that there might be an error in the provided options. Let’s double-check our work to make sure we haven’t made any mistakes.
Reviewing the Options
Let's go through the options and see which one fits both equations:
A) x = 1, y = 5
2x - y = 2(1) - 5 = 2 - 5 = -3 (This checks out for the first equation)
x/y = 1/5 which is not equal to 6/8 or 3/4 (This does not check out for the ratio)
B) x = 2, y = 7
2x - y = 2(2) - 7 = 4 - 7 = -3 (This checks out for the first equation)
x/y = 2/7 which is not equal to 6/8 or 3/4 (This does not check out for the ratio)
C) x = 3, y = 9
2x - y = 2(3) - 9 = 6 - 9 = -3 (This checks out for the first equation)
x/y = 3/9 = 1/3 which is not equal to 6/8 or 3/4 (This does not check out for the ratio)
D) x = 4, y = 11
2x - y = 2(4) - 11 = 8 - 11 = -3 (This checks out for the first equation)
x/y = 4/11 which is not equal to 6/8 or 3/4 (This does not check out for the ratio)
Reconsidering the Ratio
It looks like there might be a misunderstanding with the problem statement or the provided options. Let's focus on the first equation, 2x - y = -3, and manipulate the ratio to fit this equation. We know x/y = 3/4, so let’s express y in terms of x:
x/y = 3/4 => 4x = 3y => y = (4/3)x
Now, substitute this into the first equation:
2x - (4/3)x = -3
Multiply all terms by 3 to eliminate the fraction:
6x - 4x = -9
Combine like terms:
2x = -9
Solve for x:
x = -9/2 = -4.5
Now, find y:
y = (4/3) * (-4.5) = (4/3) * (-9/2) = -36/6 = -6
So we get x = -4.5 and y = -6 again. It seems our initial calculation was correct. Let’s analyze why none of the given options fit.
Analyzing the Discrepancy
Since none of the provided options satisfy both equations, it's possible there's an error in the question or the answer choices. However, let's consider what would happen if we tweaked the ratio slightly to see if any of the options could work.
We require that 2x - y = -3. Rearranging this, we get y = 2x + 3. So if we ensure y follows this relationship with x, then we only need to check the ratio condition against the options.
A) x = 1, y = 5: y = 2(1) + 3 = 5. So it fits the first equation. However, x/y = 1/5 is not 3/4.
B) x = 2, y = 7: y = 2(2) + 3 = 7. So it fits the first equation. However, x/y = 2/7 is not 3/4.
C) x = 3, y = 9: y = 2(3) + 3 = 9. So it fits the first equation. However, x/y = 3/9 = 1/3 is not 3/4.
D) x = 4, y = 11: y = 2(4) + 3 = 11. So it fits the first equation. However, x/y = 4/11 is not 3/4.
Conclusion
After thoroughly analyzing the problem and the given options, it appears that none of the options A, B, C, or D satisfy both equations 2x - y = -3 and x/y = 6/8. The correct solution we found through calculation is x = -4.5 and y = -6. It's possible that there was an error in the original problem statement or in the provided answer choices. If you encounter such discrepancies, it's always a good idea to double-check the problem and your calculations, and if possible, clarify with the source of the problem.
Keep practicing, guys, and you'll become math pros in no time! Remember, understanding the steps and verifying your answers is key! Great job sticking with it, and let's tackle more fun problems next time! Happy solving!