Question

А) 3x-4y=5 4x+y=8 Б)2x-5y=7 -4x+10y=3

Answer 1

А) 3х-4у=5

у=8-4х

3х-4 (8-4х)=5

х=37/19

у=8-4*37/19

у=4/19

Ответ: х=37/19., у=4/19

Б) 2х-5у=7

-4х+10у=3

х=7/2+5/2у

-4х+10у=3

-4 (7/2+5/2у)+10у=3

у€ø

Ответ:(х,у) €ø

Answer 2

To solve each system of equations, we can use either the substitution method or the elimination method. I'll demonstrate both methods for each system:

А) 3x - 4y = 5 4x + y = 8

Substitution method: Step 1: Solve one equation for one variable. From the second equation, isolate y: y = 8 - 4x

Step 2: Substitute the expression for y from Step 1 into the first equation: 3x - 4(8 - 4x) = 5

Step 3: Solve for x: 3x - 32 + 16x = 5 19x - 32 = 5 19x = 37 x = 37 / 19 x ≈ 1.947

Step 4: Now, substitute the value of x into either of the original equations to find y: 4(1.947) + y = 8 7.788 + y = 8 y = 8 - 7.788 y ≈ 0.212

So the solution to the system is approximately x ≈ 1.947 and y ≈ 0.212.

Elimination method: Step 1: Multiply both equations by suitable constants to make the coefficients of y in both equations equal. In this case, we'll multiply the first equation by 1 and the second equation by 4: 3x - 4y = 5 4(4x + y) = 4(8) 3x - 4y = 5 16x + 4y = 32

Step 2: Add the two equations to eliminate y: (3x - 4y) + (16x + 4y) = 5 + 32 19x = 37 x = 37 / 19 x ≈ 1.947

Step 3: Substitute the value of x into either of the original equations to find y: 4(1.947) + y = 8 7.788 + y = 8 y = 8 - 7.788 y ≈ 0.212

The solution is the same as before: approximately x ≈ 1.947 and y ≈ 0.212.

Now let's solve the second system:

Б) 2x - 5y = 7 -4x + 10y = 3

Substitution method: Step 1: Solve one equation for one variable. From the first equation, isolate x: x = (7 + 5y) / 2

Step 2: Substitute the expression for x from Step 1 into the second equation: -4((7 + 5y) / 2) + 10y = 3

Step 3: Solve for y: -2(7 + 5y) + 10y = 6 -14 - 10y + 10y = 6 -14 = 6 (This is a contradiction and can't be solved.)

Since the equation leads to a contradiction (false statement), there is no solution to this system of equations. The two lines representing the equations are parallel and do not intersect.

Elimination method: Step 1: Multiply both equations by suitable constants to make the coefficients of x in both equations equal. In this case, we'll multiply the first equation by 2 and the second equation by 1: 2(2x - 5y) = 2(7) -4x + 10y = 3

Step 2: Add the two equations to eliminate x: (2x - 5y) + (-4x + 10y) = 7 + 3 -5y + 10y = 10 5y = 10 y = 10 / 5 y = 2

Step 3: Substitute the value of y into either of the original equations to find x: 2x - 5(2) = 7 2x - 10 = 7 2x = 17 x = 17 / 2 x ≈ 8.5

Again, this leads to a contradiction as the system has no solution.

Therefore, the second system has no solution, and the first system has a unique solution of approximately x ≈ 1.947 and y ≈ 0.212.