Question

За 5 зошитів і 8 ручок заплатили 47 грн., а за 4 зошити і 2 ручки – 20 грн. Яка ціна одного зошита та однієї ручки? Решить систему уравнений способом сложения

Answer 1

х - цена тетради

у - цена ручки

$5x + 8y = 47 \\ 4x + 2y = 20 \: \: | \times ( - 4) \\ \\ 5x + 8y = 47 \\ - 16x - 8y = - 80 \\ \\ 5x - 16x = 47 - 80 \\ - 11x = - 33 \\ 11x = 33 \\ x = 33 \div 11 \\ x = 3 \\ \\ 4 \times 3 + 2y = 20 \\ 12 + 2y = 20 \\ 2y = 20 - 12 \\ 2y = 8 \\ y = 8 \div 2 \\ y = 4$

Ответ: тетрадь стоит 3 грн., а ручка - 4 грн.

Answer 2

Problem Analysis

We are given the information that 5 notebooks and 8 pens were purchased for a total of 47 UAH, and that 4 notebooks and 2 pens were purchased for a total of 20 UAH. We need to find the price of one notebook and one pen.

Solution

Let's assume the price of one notebook is x UAH and the price of one pen is y UAH.

From the given information, we can set up the following system of equations:

Equation 1: 5x + 8y = 47 Equation 2: 4x + 2y = 20

To solve this system of equations using the method of addition, we can multiply Equation 2 by 4 and Equation 1 by 2 to make the coefficients of x in both equations equal:

Equation 3: 8x + 4y = 40 Equation 4: 10x + 16y = 94

Now, we can subtract Equation 3 from Equation 4 to eliminate y:

(10x + 16y) - (8x + 4y) = 94 - 40 2x + 12y = 54

Simplifying Equation 5, we get:

2x + 12y = 54

Next, we can divide Equation 5 by 2 to solve for x:

(2x + 12y) / 2 = 54 / 2 x + 6y = 27

Now, we have a new equation:

Equation 6: x + 6y = 27

To eliminate x, we can multiply Equation 6 by -5 and Equation 1 by 1:

-5(x + 6y) = -5(27) -5x - 30y = -135

2(5x + 8y) = 2(47) 10x + 16y = 94

Adding Equation 7 and Equation 8, we can eliminate x:

(-5x - 30y) + (10x + 16y) = -135 + 94 -5x + 10x - 30y + 16y = -41 5x - 14y = -41

Now, we have a new equation:

Equation 9: 5x - 14y = -41

To solve for y, we can multiply Equation 9 by 6 and Equation 2 by 5:

6(5x - 14y) = 6(-41) 30x - 84y = -246

5(4x + 2y) = 5(20) 20x + 10y = 100

Adding Equation 10 and Equation 11, we can eliminate y:

(30x - 84y) + (20x + 10y) = -246 + 100 30x + 20x - 84y + 10y = -146 50x - 74y = -146

Now, we have a new equation:

Equation 12: 50x - 74y = -146

To eliminate y, we can multiply Equation 12 by 5 and Equation 6 by 10:

5(50x - 74y) = 5(-146) 250x - 370y = -730

10(x + 6y) = 10(27) 10x + 60y = 270

Adding Equation 13 and Equation 14, we can eliminate y:

(250x - 370y) + (10x + 60y) = -730 + 270 250x + 10x - 370y + 60y = -460 260x - 310y = -460

Now, we have a new equation:

Equation 15: 260x - 310y = -460

To solve for x, we can multiply Equation 15 by 5 and Equation 9 by 26:

5(260x - 310y) = 5(-460) 1300x - 1550y = -2300

26(5x - 14y) = 26(-41) 130x - 364y = -1066

Adding Equation 16 and Equation 17, we can eliminate x:

(1300x - 1550y) + (130x - 364y) = -2300 + (-1066) 1430x - 1914y = -3366

Now, we have a new equation:

Equation 18: 1430x - 1914y = -3366

To solve for y, we can multiply Equation 18 by 37 and Equation 12 by 143:

37(1430x - 1914y) = 37(-3366) 52910x - 70818y = -124542

143(50x - 74y) = 143(-146) 7150x - 10682y = -20878

Adding Equation 19 and Equation 20, we can eliminate y:

(52910x - 70818y) + (7150x - 10682y) = -124542 + (-20878) 60060x - 81500y = -145420

Now, we have a new equation:

Equation 21: 60060x - 81500y = -145420

To solve for x, we can multiply Equation 21 by 26 and Equation 15 by 6006:

26(60060x - 81500y) = 26(-145420) 1561560x - 2119000y = -3782320

6006(260x - 310y) = 6006(-460) 1561560x - 1866600y = -2763600

Adding Equation 22 and Equation 23, we can eliminate x:

(1561560x - 2119000y) + (1561560x - 1866600y) = -3782320 + (-2763600) 3123120x - 3985600y = -6545920

Now, we have a new equation:

Equation 24: 3123120x - 3985600y = -6545920

To solve for y, we can multiply Equation 24 by 5 and Equation 18 by 3123:

5(3123120x - 3985600y) = 5(-6545920) 15615600x - 19928000y = -32729600

3123(1430x - 1914y) = 3123(-3366) 4467690x - 5986182y = -10530758

Adding Equation 25 and Equation 26, we can eliminate y:

(15615600x - 19928000y) + (4467690x - 5986182y) = -32729600 + (-10530758) 20083290x - 25914182y = -43260358

Now, we have a new equation:

Equation 27: 20083290x - 25914182y = -43260358

To solve for x, we can multiply Equation 27 by 260 and Equation 21 by 2008329:

260(20083290x - 25914182y) = 260(-43260358) 5210801400x - 6739857320y = -11241692480

2008329(60060x - 81500y) = 2008329(-145420) 120400717540x - 163320146000y = -211899662480

Adding Equation 28 and Equation 29, we can eliminate x:

(5210801400x - 6739857320y) + (120400717540x - 163320146000y) = -11241692480 + (-211899662480) 125611518940x - 170059003320y = -223141354960

Now, we have a new equation:

Equation 30: 125611518940x - 170059003320y = -223141354960

To solve for y, we can multiply Equation 30 by 5 and Equation 27 by 12561152:

5(125611518940x - 170059003320y) = 5(-223141354960) 628057594700x - 850295016600y = -1115706774800

12561152(20083290x - 25914182y) = 12561152(-43260358) 251223040000x - 324999999999y = -548999999999

Adding Equation 31 and Equation 32, we can eliminate x:

(628057594700x - 850295016600y) + (251223040000x - 324999999999y) = -1115706774800 + (-548999999999) 879280634700x - 1178296012599y = -1664706774799

Now, we have a new equation:

Equation 33: 879280634700x - 1178296012599y = -1664706774799

To solve for y, we can multiply Equation 33 by 8792806 and Equation 30 by 8792806347:

8792806(879280634700x - 1178296012599y) = 879