Question

1280. 1) [10(x + y) = 30 + 8x, 9(x - y) = -49-8y; Y 2) (21(x - y) = 48 + 20x, 19(y + x) = 100 + 12y.​

Answer 1

1)

$\displaystyle\bf\\\left \{ {{10(x + y) = 30 + 8x} \atop {9(x - y) = - 49 - 8y }} \right. \\ \displaystyle\bf\\\left \{ {{10x + 10y - 8x = 30} \atop {9x - 9y + 8y = - 49 }} \right. \\ \displaystyle\bf\\\left \{ {{2x + 10y = 30} \atop {9x - y = - 49 }} \right. \\ \displaystyle\bf\\\left \{ {{x = 15 - 5y} \atop {9(15 - 5y) - y = - 49 }} \right. \\ \\ 135 - 45y - y = - 49 \\ - 46y = - 49 - 135 \\ 46y = 184 \\ y = 184 \div 46 \\ y = 4 \\ \\ x = 15 - 5 \times 4 \\ x = 15 - 20 \\ x = - 5$

Ответ: ( - 5 ; 4 )

2)

$\displaystyle\bf\\\left \{ {{21(x - y) = 48 + 20x} \atop {19(y + x) = 100 + 12y }} \right. \\ \displaystyle\bf\\\left \{ {{21x - 21y - 20x = 48 } \atop {19x + 19y - 12y = 100 }} \right. \\ \displaystyle\bf\\\left \{ {{x - 21y = 48} \atop {19x + 7y = 100 }} \right. \\ \displaystyle\bf\\\left \{ {{x = 21y + 48} \atop {19(21y + 48) +7 y = 100 }} \right. \\ \\ 399y + 912 + 7y = 100 \\ 406y = 100 - 912 \\ 406y = - 812 \\ y = - 812 \div 406 \\ y = - 2 \\ \\ x = 21 \times ( - 2) + 48 \\ x = - 42 + 48 \\ x = 6$

Ответ: ( 6 ; - 2 )

Answer 2

Пошаговое объяснение:

№1

{10(х + у) = 30 + 8х

{9(х - у) = -49 - 8у

{10х + 10у = 30 + 8х

{9х - 9у = -49 - 8у

{10х + 10у - 8х = 30

{9х - 9у + 8у = -49

{2х + 10у = 30

{9х - у = -49

{2х + 10у = 30

{у = 9х + 49

1)

2х + 10у = 30

2х + 10(9х + 49) = 30

2х + 90х + 490 = 30

92х = 30 - 490

92х = -460

х = -460 : 92

х = -5

2)

у = 9х + 49

у = 9 * (-5) + 49

у = -45 + 49

у = 4

(-5; 4)

№2

{21(х - у) = 48 + 20х

{19(у + х) = 100 + 12у

{21х - 21у = 48 + 20х

{19у + 19х = 100 + 12у

{21х - 21у - 20х = 48

{19у + 19х - 12у = 100

{х - 21у = 48

{19х + 7у = 100

{х = 48 + 21у

{19х + 7у = 100

1)

19х + 7у = 100

19(48 + 21у) + 7у = 100

912 + 399у + 7у = 100

399у + 7у = 100 - 912

406у = -812

у = -812 : 406

у = -2

2)

х = 48 + 21у

х = 48 + 21 * (-2)

х = 48 - 42

х = 6

(6; -2)

Answer 3

Let's solve these systems of linear equations step by step.

1. [10(x + y) = 30 + 8x, 9(x - y) = -49 - 8y]

First, expand the equations:

10x + 10y = 30 + 8x (Equation 1) 9x - 9y = -49 - 8y (Equation 2)

Now, simplify the equations:

10x - 8x + 10y = 30 (Combine like terms in Equation 1) 9x + 8y - 9y = -49 (Combine like terms in Equation 2)

2x + 10y = 30 (Simplify Equation 1) 9x - y = -49 (Simplify Equation 2)

Now, let's solve this system of equations using the method of substitution or elimination. We'll use the elimination method.

Multiply Equation 2 by 10 to make the coefficients of y in both equations equal:

20x + 100y = 300 (Multiply Equation 1 by 10) 90x - 10y = -490 (Multiply Equation 2 by 10)

Now, add the two equations together to eliminate y:

(20x + 100y) + (90x - 10y) = 300 - 490

Combine like terms:

110x + 90y = -190

Now, let's solve for x:

110x = -190 - 90y

Divide both sides by 110:

x = (-190 - 90y) / 110

Now, let's solve for y:

2x + 10y = 30

2[(-190 - 90y) / 110] + 10y = 30

Multiply both sides by 110 to get rid of the fraction:

2(-190 - 90y) + 110(10y) = 3300

Now, simplify:

-380 - 180y + 1100y = 3300

Combine like terms:

-380 + 920y = 3300

Add 380 to both sides:

920y = 3300 + 380

920y = 3680

Now, divide by 920 to solve for y:

y = 3680 / 920 y = 4

Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use Equation 1:

2x + 10y = 30

2x + 10(4) = 30

2x + 40 = 30

Subtract 40 from both sides:

2x = 30 - 40 2x = -10

Now, divide by 2 to solve for x:

x = -10 / 2 x = -5

So, the solution to the first system of equations is x = -5 and y = 4.

Now, let's move on to the second system of equations:

2. [21(x - y) = 48 + 20x, 19(y + x) = 100 + 12y]

First, expand the equations:

21x - 21y = 48 + 20x (Equation 1) 19y + 19x = 100 + 12y (Equation 2)

Now, simplify the equations:

21x - 20x - 21y = 48 (Combine like terms in Equation 1) 19x - 12y - 19y = 100 (Combine like terms in Equation 2)

x - 21y = 48 (Simplify Equation 1) 19x - 31y = 100 (Simplify Equation 2)

Now, let's solve this system of equations using the elimination method. Multiply Equation 1 by 19 and Equation 2 by 21 to make the coefficients of x in both equations equal:

19(x - 21y) = 19(48) (Multiply Equation 1 by 19) 21(19x - 31y) = 21(100) (Multiply Equation 2 by 21)

Now, simplify:

19x - 399y = 912 (Simplify Equation 1) 399x - 651y = 2100 (Simplify Equation 2)

Now, add the two equations together to eliminate x:

(19x - 399y) + (399x - 651y) = 912 + 2100

Combine like terms:

418x - 1050y = 3012

Now, let's solve for x:

418x = 3012 + 1050y

Divide both sides by 418:

x = (3012 + 1050y) / 418

Now, we can substitute this expression for x into Equation 1:

x - 21y = 48

[(3012 + 1050y) / 418] - 21y = 48

Now, simplify:

(3012 + 1050y) - 21y418 = 48418

Expand:

3012 + 1050y - 8788y = 20064

Combine like terms:

-7738y = 20064 - 3012

-7738y = 17052

Now, divide by -7738 to solve for y:

y = 17052 / -7738 y ≈ -2.2035 (rounded to four decimal places)

Now that we have the value of y, we can substitute it back into the expression we found for x:

x = (3012 + 1050y) / 418

x = (3012 + 1050*(-2.2035)) / 418

x ≈ (3012 - 2315.617) / 418

x ≈ 6.7101 (rounded to four decimal places)

So, the solution to the second system of equations is x ≈ 6.7101 and y ≈ -2.2035 (rounded to four decimal places).