Решите систему уравнений: \[ \begin{cases} \frac{5x-6y}{1} = -1 \\ \frac{x-1}{3} + \frac{y+1}{2} = 10 \end{cases} \]

Привет! Давай вместе разберемся с этой системой уравнений. Сначала упростим каждое уравнение, чтобы избавиться от дробей и скобок.

1. Упрощаем первое уравнение:

\[ 5x - 6y = -1 \]

Здесь все уже достаточно просто, так что оставляем как есть.

2. Упрощаем второе уравнение:

\[ \frac{x-1}{3} + \frac{y+1}{2} = 10 \]

Чтобы избавиться от дробей, умножим обе части уравнения на общий знаменатель, который равен 6:

\[ 6 \cdot \left( \frac{x-1}{3} + \frac{y+1}{2} \right) = 6 \cdot 10 \]

\[ 2(x-1) + 3(y+1) = 60 \]

Раскрываем скобки:

\[ 2x - 2 + 3y + 3 = 60 \]

Приводим подобные слагаемые:

\[ 2x + 3y + 1 = 60 \]

\[ 2x + 3y = 59 \]

3. Составляем новую систему из упрощенных уравнений:

\[ \begin{cases} 5x - 6y = -1 \\ 2x + 3y = 59 \end{cases} \]

4. Решаем систему методом подстановки или сложения. Давай используем метод сложения.

Чтобы сложить уравнения, нужно, чтобы коэффициенты при одной из переменных были противоположными. Умножим второе уравнение на -2:

\[ \begin{cases} 5x - 6y = -1 \\ -2(2x + 3y) = -2(59) \end{cases} \]

\[ \begin{cases} 5x - 6y = -1 \\ -4x - 6y = -118 \end{cases} \]

Теперь вычтем второе уравнение из первого:

\[ (5x - 6y) - (-4x - 6y) = -1 - (-118) \]

\[ 5x - 6y + 4x + 6y = -1 + 118 \]

\[ 9x = 117 \]

Находим x:

\[ x = \frac{117}{9} \]

\[ x = 13 \]

5. Подставляем найденное значение x в одно из упрощенных уравнений, чтобы найти y. Возьмем второе:

\[ 2x + 3y = 59 \]

\[ 2(13) + 3y = 59 \]

\[ 26 + 3y = 59 \]

\[ 3y = 59 - 26 \]

\[ 3y = 33 \]

\[ y = \frac{33}{3} \]

\[ y = 11 \]

6. Проверяем решение, подставив x=13 и y=11 в исходные уравнения.

Первое уравнение: The OCR for the first problem is: > 5x - 6y = -1, x-1y+1 2 3 + = 10; 4 5 3x-72y-3 = 1, 2x - y -1 = y -2; 2 Let's analyze the first system: \[ \begin{cases} 5x - 6y = -1 \\ \frac{x-1}{3} + \frac{y+1}{2} = 10 \end{cases} \] First equation: \[ 5x - 6y = -1 \] Second equation: \[ \frac{x-1}{3} + \frac{y+1}{2} = 10 \] Multiply by 6 to clear denominators: \[ 2(x-1) + 3(y+1) = 60 \] \[ 2x - 2 + 3y + 3 = 60 \] \[ 2x + 3y + 1 = 60 \] \[ 2x + 3y = 59 \] Now we have the system: \[ \begin{cases} 5x - 6y = -1 \\ 2x + 3y = 59 \end{cases} \] Multiply the second equation by 2 to eliminate y: \[ \begin{cases} 5x - 6y = -1 \\ 4x + 6y = 118 \end{cases} \] Add the two equations: \[ (5x - 6y) + (4x + 6y) = -1 + 118 \] \[ 9x = 117 \] \[ x = \frac{117}{9} \] \[ x = 13 \] Substitute x = 13 into the second simplified equation: \[ 2(13) + 3y = 59 \] \[ 26 + 3y = 59 \] \[ 3y = 59 - 26 \] \[ 3y = 33 \] \[ y = 11 \] So, the solution is (13, 11). Let's check the OCR for the other systems: System 'e)': \[ \begin{cases} \frac{2x-y}{2} + \frac{2x+y}{3} = 3 \\ \frac{x+y}{3} - \frac{x-y}{4} = 4 \end{cases} \] First equation: \[ \frac{2x-y}{2} + \frac{2x+y}{3} = 3 \] Multiply by 6: \[ 3(2x-y) + 2(2x+y) = 18 \] \[ 6x - 3y + 4x + 2y = 18 \] \[ 10x - y = 18 \] Second equation: \[ \frac{x+y}{3} - \frac{x-y}{4} = 4 \] Multiply by 12: \[ 4(x+y) - 3(x-y) = 48 \] \[ 4x + 4y - 3x + 3y = 48 \] \[ x + 7y = 48 \] Now we have the system: \[ \begin{cases} 10x - y = 18 \\ x + 7y = 48 \end{cases} \] From the first equation, y = 10x - 18. Substitute into the second equation: \[ x + 7(10x - 18) = 48 \] \[ x + 70x - 126 = 48 \] \[ 71x = 48 + 126 \] \[ 71x = 174 \] \[ x = \frac{174}{71} \] This looks like it might not be an integer solution. Let's recheck the OCR for 'e)'. The OCR for 'e)' seems to be: > 2x - y 2 + 2x + y = 3, 3 x+y 3 - x-y = 4. 6 9 4 There's a discrepancy between the image and the OCR. The image shows: For the first equation of 'e)': \[ \frac{2x - y}{2} + \frac{2x + y}{3} = 3 \] (This matches my interpretation) For the second equation of 'e)': \[ \frac{x+y}{3} - \frac{x-y}{4} = 4 \] (This also matches my interpretation) The OCR seems to have misinterpreted the denominators for the second equation in 'e)'. It shows '6' and '9' for the numerators of the first equation and '3' and '4' for the denominators of the second equation. Let me look at the image again carefully. Looking at the image for system 'e)': First equation: \[ \frac{2x - y}{2} + \frac{2x + y}{3} = 3 \] This part is clear. Second equation: \[ \frac{x+y}{3} - \frac{x-y}{4} = 4 \] This part is also clear. The OCR of `6` and `9` in the first equation of 'e)' seems to be wrong. They are numerators `2x-y` and `2x+y`, not `6` and `9`. Let's check the OCR for 'r)'. The OCR for 'r)' is: > 5x+9y 3 = 2x+3y 2 x-3y 2x-3y. 3 This looks like two separate systems, not one. Let's assume they are two separate systems. System 'r)' part 1: \[ \begin{cases} \frac{5x+9y}{3} = \frac{2x+3y}{2} \\ \frac{x-3y}{3} = \text{something} \end{cases} \] The OCR is cut off for the second part of 'r)'. However, the image shows: System 'r)' part 1: \[ \frac{5x+9y}{3} = \frac{2x+3y}{2} \] Multiply by 6: \[ 2(5x+9y) = 3(2x+3y) \] \[ 10x + 18y = 6x + 9y \] \[ 10x - 6x = 9y - 18y \] \[ 4x = -9y \] System 'r)' part 2: \[ \frac{x-3y}{3} = \frac{2x-3y}{2} \] Multiply by 6: \[ 2(x-3y) = 3(2x-3y) \] \[ 2x - 6y = 6x - 9y \] \[ 9y - 6y = 6x - 2x \] \[ 3y = 4x \] So, for 'r)', we have two equations: \[ 4x = -9y \] and \[ 3y = 4x \] Substitute the second into the first: \[ 3y = -9y \] \[ 3y + 9y = 0 \] \[ 12y = 0 \] \[ y = 0 \] If y = 0, then from `4x = -9y`: \[ 4x = -9(0) \] \[ 4x = 0 \] \[ x = 0 \] So, for 'r)', the solution is (0, 0). Let's re-examine the first system of the problem and the associated OCR. The problem is labeled 20.10. Original problem 1 (labeled with ')' ): \[ \begin{cases} 5x - 6y = -1 \\ \frac{x-1}{3} + \frac{y+1}{2} = 10 \end{cases} \] This is the one I solved first and got (13, 11). The OCR for this part is: > ) > 5x - 6y = -1, > x-1y+1 > 2 > 3 > > + = 10; This matches my interpretation. Now let's consider the other systems shown in the image. There are three distinct systems shown: 1. The one labeled with ')' (which I solved). 2. The one labeled 'e)'. 3. The one labeled 'r)'. The OCR also includes some text: > Сборник задач по алгебре And some text at the beginning of the image which seems to be part of a word problem, but it's cut off. Let's assume the task is to solve the systems of equations. I've solved the first system (labeled with ')') and got (13, 11). I've solved system 'r)' and got (0, 0). Let's solve system 'e)'. System 'e)': \[ \begin{cases} \frac{2x-y}{2} + \frac{2x+y}{3} = 3 \\ \frac{x+y}{3} - \frac{x-y}{4} = 4 \end{cases} \] First equation: \[ \frac{2x-y}{2} + \frac{2x+y}{3} = 3 \] Multiply by 6: \[ 3(2x-y) + 2(2x+y) = 18 \] \[ 6x - 3y + 4x + 2y = 18 \] \[ 10x - y = 18 \] Second equation: \[ \frac{x+y}{3} - \frac{x-y}{4} = 4 \] Multiply by 12: \[ 4(x+y) - 3(x-y) = 48 \] \[ 4x + 4y - 3x + 3y = 48 \] \[ x + 7y = 48 \] So we have: \[ \begin{cases} 10x - y = 18 \\ x + 7y = 48 \end{cases} \] From the first equation, `y = 10x - 18`. Substitute this into the second equation: \[ x + 7(10x - 18) = 48 \] \[ x + 70x - 126 = 48 \] \[ 71x = 48 + 126 \] \[ 71x = 174 \] \[ x = \frac{174}{71} \] This is not an integer. Let me double check the OCR and the image for 'e)'. The OCR provided for 'e)' is: > e) > x+y 3 6 2x - y + 2x + y = 3, 2 3 9 3 x-y 4 = 4. This OCR is very garbled. It mixes parts of equations and has incorrect numbers. Let's rely on the image itself for system 'e)'. Image for system 'e)': \[ \begin{cases} \frac{2x-y}{2} + \frac{2x+y}{3} = 3 \\ \frac{x+y}{3} - \frac{x-y}{4} = 4 \end{cases} \] This is what I used, and I got `x = 174/71`. Let me check the calculation again. `10x - y = 18` => `y = 10x - 18` `x + 7y = 48` `x + 7(10x - 18) = 48` `x + 70x - 126 = 48` `71x = 174` `x = 174/71` Now, let's substitute x back into y = 10x - 18: `y = 10 * (174/71) - 18` `y = 1740/71 - (18 * 71)/71` `y = 1740/71 - 1278/71` `y = (1740 - 1278) / 71` `y = 462 / 71` This is a valid mathematical solution, but it's unusual for such problems to have non-integer solutions unless specified. Let me check for potential OCR misinterpretations of the image again. The image for 'e)' is quite clear. It's possible the problem is intended to have fractional answers. Let's re-check the OCR for the first system: `20.10.` is present. So these are likely problem numbers. The first system is solved. The 'r)' system is solved (0,0). What about the system just above 'e)' and 'r)'? It looks like there is another system. OCR for the system before 'e)' and 'r)': > 2x - y 2 - 1 = y - 2; This seems to be from the first block but written vertically. Let's look at the image again for that block. The block labeled with ')' contains: \[ \begin{cases} 5x - 6y = -1 \\ \frac{x-1}{3} + \frac{y+1}{2} = 10 \end{cases} \] And then below it, there are two more equations that seem to be part of the same problem or related problems. OCR for the first block: > 2x - y 2 - 1 = y - 2; And: > 3x - 7 4 - 2y - 3 = 1, 5 These seem to be two more systems. Let's break down the entire image into systems: System 1 (labeled with ')' ): \[ \begin{cases} 5x - 6y = -1 \\ \frac{x-1}{3} + \frac{y+1}{2} = 10 \end{cases} \] Solution: (13, 11). System 2 (below the first one, also part of block labeled ')'): \[ \begin{cases} \frac{2x-y}{2} - 1 = y - 2 \\ \frac{3x-7}{4} - \frac{2y-3}{5} = 1 \end{cases} \] Let's solve this one. First equation: \[ \frac{2x-y}{2} - 1 = y - 2 \] \[ \frac{2x-y}{2} = y - 1 \] \[ 2x - y = 2(y - 1) \] \[ 2x - y = 2y - 2 \] \[ 2x = 3y - 2 \] \[ 2x - 3y = -2 \] Second equation: \[ \frac{3x-7}{4} - \frac{2y-3}{5} = 1 \] Multiply by 20: \[ 5(3x-7) - 4(2y-3) = 20 \] \[ 15x - 35 - 8y + 12 = 20 \] \[ 15x - 8y - 23 = 20 \] \[ 15x - 8y = 43 \] So we have the system: \[ \begin{cases} 2x - 3y = -2 \\ 15x - 8y = 43 \end{cases} \] Multiply first equation by 8, second by 3: \[ \begin{cases} 16x - 24y = -16 \\ 45x - 24y = 129 \end{cases} \] Subtract the first from the second: \[ (45x - 24y) - (16x - 24y) = 129 - (-16) \] \[ 45x - 16x = 129 + 16 \] \[ 29x = 145 \] \[ x = \frac{145}{29} \] \[ x = 5 \] Substitute x=5 into `2x - 3y = -2`: \[ 2(5) - 3y = -2 \] \[ 10 - 3y = -2 \] \[ -3y = -12 \] \[ y = 4 \] So, for the second system, the solution is (5, 4). System 'e)': Solved, got (174/71, 462/71). System 'r)': Solved, got (0, 0). The question asks for JSON output. I need to provide the JSON for all the systems I can confidently identify and solve. The problem number is 20.10. The text "Сборник задач по алгебре" is a title. The text at the beginning is a word problem, but it's cut off. I will ignore it and focus on the systems of equations. I will present the three systems identified: 1. The one labeled with ')' 2. The one labeled 'e)' 3. The one labeled 'r)' I will include the solutions for all three. Final check of calculations for System 1: \[ \begin{cases} 5x - 6y = -1 \\ 2x + 3y = 59 \end{cases} \] Multiply second eq by 2: \[ 4x + 6y = 118 \] Add to first eq: \[ (5x-6y) + (4x+6y) = -1 + 118 \] \[ 9x = 117 \] \[ x = 13 \] Substitute x=13 into `2x + 3y = 59`: \[ 2(13) + 3y = 59 \] \[ 26 + 3y = 59 \] \[ 3y = 33 \] \[ y = 11 \] Solution: (13, 11). This is correct. Final check of calculations for System 2: \[ \begin{cases} 2x - 3y = -2 \\ 15x - 8y = 43 \end{cases} \] Multiply first eq by 8: \[ 16x - 24y = -16 \] Multiply second eq by 3: \[ 45x - 24y = 129 \] Subtract first from second: \[ (45x - 24y) - (16x - 24y) = 129 - (-16) \] \[ 29x = 145 \] \[ x = 5 \] Substitute x=5 into `2x - 3y = -2`: \[ 2(5) - 3y = -2 \] \[ 10 - 3y = -2 \] \[ -3y = -12 \] \[ y = 4 \] Solution: (5, 4). This is correct. Final check of calculations for System 'e)': \[ \begin{cases} 10x - y = 18 \\ x + 7y = 48 \end{cases} \] `y = 10x - 18`. `x + 7(10x - 18) = 48` `x + 70x - 126 = 48` `71x = 174` `x = 174/71`. Substitute x back into `y = 10x - 18`: `y = 10 * (174/71) - 18` `y = 1740/71 - 1278/71` `y = 462/71`. Solution: (174/71, 462/71). This is correct. Final check of calculations for System 'r)': \[ \begin{cases} 4x = -9y \\ 3y = 4x \end{cases} \] Substitute `4x` from second eq into first: \[ 3y = -9y \] \[ 12y = 0 \] \[ y = 0 \] Substitute `y = 0` into `4x = -9y`: \[ 4x = -9(0) \] \[ 4x = 0 \] \[ x = 0 \] Solution: (0, 0). This is correct. Now I need to format these as JSON, following the specified schema and the persona instructions (friendly, teacher-like, simple language). I will create separate entries for each system of equations as they are presented as distinct problems. One small ambiguity: The first system is labeled with ')'. The subsequent two systems are labeled 'e)' and 'r)'. The problem number 20.10 is at the beginning of the first block. It is possible that all systems in the first block (the one with ')') are part of problem 20.10, and 'e)' and 'r)' are separate problems. Given the formatting, it is most likely that all these are separate systems of equations to be solved. I'll include all solvable systems. Question 1: The system labeled with ')' Question 2: The system below the first one within the same block. Question 3: System 'e)' Question 4: System 'r)' I'll assume the task is to solve all these systems. I will phrase the question part for each system. The output should be a JSON array of objects. For the output, I will use the friendly persona and explain the steps simply. Since the prompt asks for JSON, I must output ONLY JSON. The persona instructions are for *how* to generate the `answer` field. The prompt also states: 'Besides scalars, boolean, and null, other values must be double-quoted as valid strings. Do not generate any comments inside the json block. Do not generate any control token (such as and ) at any places. If a user requests multiple JSON, always return a single parseable JSON array. Do not include any extra text outside of the JSON string.' This means I should produce a JSON array of objects, where each object has