Question

1)9x-10y=5 2)3y-9x=5.6​

Answer 1

$1)9x - 10y = 5 \\ 9x - 10y - 5 = 0 \\ f(x,y) = 9x - 10y - 5 \\ f_{x} = 9 \\ f_{y} = - 10 \\ \frac{dy}{dx} = - \frac{9}{ - 10 } \\ \frac{dy}{dx} = \frac{9}{10}$

$2)3y - 9x = 5.6 \\ 3y - 9x - 5.6 = 0 \\ f(x,y) = 3y - 9x - 5.6 \\ f_{x} = - 9 \\ f_{y} = 3 \\ \frac{dy}{dx} = - \frac{ - 9}{3} \\ \frac{dy}{dx} = - ( - 3) \\ \frac{dy}{dx} = 3$

Answer 2

To solve this system of linear equations:

1. 9x - 10y = 5
2. 3y - 9x = 5.6

You can use the method of substitution or elimination. Let's use the elimination method to solve this system.

First, we need to make the coefficients of either x or y in both equations equal. In this case, we can make the coefficients of x equal by multiplying the second equation by 3:

1. 9x - 10y = 5
2. 9x - 3(3y) = 3(5.6)

Now, the system becomes:

1. 9x - 10y = 5
2. 9x - 9y = 16.8

Now, subtract equation (2) from equation (1) to eliminate x:

(9x - 10y) - (9x - 9y) = 5 - 16.8

This simplifies to:

-10y + 9y = -11.8

Now, combine like terms:

-y = -11.8

Divide both sides by -1 to isolate y:

y = 11.8

Now that you have found the value of y, you can substitute it back into either equation (1) or (2) to solve for x. Let's use equation (1):

9x - 10(11.8) = 5

9x - 118 = 5

Add 118 to both sides:

9x = 5 + 118

9x = 123

Divide both sides by 9:

x = 123 / 9

x = 13.67 (rounded to two decimal places)

So, the solution to the system of equations is: x ≈ 13.67 y ≈ 11.8