Solving A System Of Equations: 4x - Y = 7, 2xy + Y = 35

Hey guys! Today, we're diving into a fun math problem: solving a system of equations. Specifically, we need to find the values of x and y that satisfy both of these equations:

1. 4x - y = 7
2. 2xy + y = 35

It might look a bit intimidating at first, but don't worry! We'll break it down step by step. Systems of equations pop up everywhere, from simple algebra problems to complex engineering designs, so mastering this is super useful. Let's get started!

Step 1: Isolating a Variable

The first thing we want to do is isolate one of the variables in one of the equations. Looking at the first equation, 4x - y = 7, it seems easiest to isolate y. We can do this by adding y to both sides and subtracting 7 from both sides. This gives us:

y = 4x - 7

Now we have y expressed in terms of x. This is a crucial step because it allows us to substitute this expression into the second equation. Isolating a variable simplifies the problem and sets us up for the next phase of solving the system.

Step 2: Substitution

Now that we have y = 4x - 7, we can substitute this into the second equation, 2xy + y = 35. Wherever we see y in the second equation, we'll replace it with (4x - 7). This gives us:

2x(4x - 7) + (4x - 7) = 35

Expanding this, we get:

8x² - 14x + 4x - 7 = 35

Combining like terms, we have:

8x² - 10x - 7 = 35

To solve for x, we need to set the equation equal to zero. Subtract 35 from both sides:

8x² - 10x - 42 = 0

Now we have a quadratic equation that we can solve. Substitution is a powerful technique because it reduces a system of two equations into a single equation with one variable, making it much easier to handle. Make sure to double-check your work during the substitution to avoid errors!

Step 3: Solving the Quadratic Equation

Okay, so we've got the quadratic equation 8x² - 10x - 42 = 0. To make things a bit easier, let's first simplify it by dividing the entire equation by 2:

4x² - 5x - 21 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's try factoring first. We are looking for two numbers that multiply to (4 * -21 = -84) and add up to -5. Those numbers are -12 and 7. So we can rewrite the middle term:

4x² - 12x + 7x - 21 = 0

Now, factor by grouping:

4x(x - 3) + 7(x - 3) = 0

(x - 3)(4x + 7) = 0

Setting each factor equal to zero gives us:

x - 3 = 0 => x = 3

4x + 7 = 0 => 4x = -7 => x = -7/4

So we have two possible values for x: 3 and -7/4. Factoring is often the quickest method if you can spot the factors easily. If not, the quadratic formula is always a reliable alternative. Remember, the quadratic formula is:

x = [-b ± sqrt(b² - 4ac)] / (2a)

In our simplified equation 4x² - 5x - 21 = 0, a = 4, b = -5, and c = -21. Plugging these values into the quadratic formula will give you the same solutions for x.

Step 4: Finding the Corresponding Values of y

Now that we have the values of x, we need to find the corresponding values of y. We can use the equation we found earlier, y = 4x - 7.

For x = 3:

y = 4(3) - 7 = 12 - 7 = 5

So one solution is (x, y) = (3, 5).

For x = -7/4:

y = 4(-7/4) - 7 = -7 - 7 = -14

So another solution is (x, y) = (-7/4, -14).

Therefore, we have two solutions for the system of equations: (3, 5) and (-7/4, -14). Always remember to plug your solutions back into the original equations to verify that they are correct. This helps catch any potential errors made during the solving process.

Step 5: Verification

To make sure our solutions are correct, let's plug them back into the original equations:

For (3, 5):

1. 4x - y = 7 => 4(3) - 5 = 12 - 5 = 7 (Correct)
2. 2xy + y = 35 => 2(3)(5) + 5 = 30 + 5 = 35 (Correct)

For (-7/4, -14):

1. 4x - y = 7 => 4(-7/4) - (-14) = -7 + 14 = 7 (Correct)
2. 2xy + y = 35 => 2(-7/4)(-14) + (-14) = (7/2)(14) - 14 = 49 - 14 = 35 (Correct)

Both solutions satisfy both equations, so they are indeed correct! Verification is a vital step to ensure accuracy, especially in more complex problems. It’s like the final seal of approval on your hard work.

Conclusion

Alright, guys! We successfully solved the system of equations:

1. 4x - y = 7
2. 2xy + y = 35

We found two solutions: (3, 5) and (-7/4, -14). Remember, the key steps were isolating a variable, substituting, solving the resulting equation (in this case, a quadratic), and verifying the solutions. Practice makes perfect, so keep solving more systems of equations to sharpen your skills.

Understanding how to solve systems of equations is fundamental in mathematics and has numerous applications in various fields. Whether you're working on physics problems, economic models, or computer simulations, the ability to solve systems of equations will undoubtedly come in handy. So, keep honing your skills and exploring the fascinating world of mathematics! You got this!