Question

3xy + x = 8,3xy + y = 7.​

Answer 1

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Answer 2

To solve the given system of equations:

Equation 1: 3xy + x = 8 Equation 2: 3xy + y = 7

We can solve this system by using the method of substitution.

Let's solve Equation 1 for x: 3xy + x = 8 x(3y + 1) = 8 x = 8 / (3y + 1)

Now substitute this value of x into Equation 2: 3xy + y = 7 3(8 / (3y + 1))y + y = 7 24y / (3y + 1) + y = 7

To simplify this expression, we need a common denominator. Multiply the y term by (3y + 1)/(3y + 1): (24y + y(3y + 1)) / (3y + 1) = 7 (24y + 3y^2 + y) / (3y + 1) = 7 (3y^2 + 25y) / (3y + 1) = 7

Multiply both sides of the equation by (3y + 1): 3y^2 + 25y = 7(3y + 1) 3y^2 + 25y = 21y + 7 3y^2 + 4y - 7 = 0

Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a)

For our quadratic equation, a = 3, b = 4, and c = -7. Plugging in these values, we get:

y = (-4 ± √(4^2 - 4 * 3 * -7)) / (2 * 3) y = (-4 ± √(16 + 84)) / 6 y = (-4 ± √100) / 6 y = (-4 ± 10) / 6

Now we have two possible values for y:

1. When y = (-4 + 10) / 6 = 6 / 6 = 1
2. When y = (-4 - 10) / 6 = -14 / 6 = -7 / 3

Now substitute these values back into Equation 1 to find the corresponding values of x:

For y = 1: x = 8 / (3 * 1 + 1) = 8 / 4 = 2

For y = -7 / 3: x = 8 / (3 * (-7 / 3) + 1) = 8 / (-7 + 1) = 8 / -6 = -4 / 3

Therefore, the solution to the system of equations is: (x, y) = (2, 1) and (x, y) = (-4/3, -7/3)