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МатематикаВопрос:
Solve the system of equations: { 6z - 5x = 2, 4z - 2x = 10
Ответ:
The system of equations is:
- 6z - 5x = 2
- 4z - 2x = 10
Insight: We can solve this system using the elimination method. To make the coefficients of 'z' or 'x' easier to work with, we can find a common multiple. Multiplying the second equation by 3 and the first equation by 2 might be helpful for eliminating 'z'. Alternatively, we can manipulate the equations to eliminate 'x'. Let's aim to eliminate 'x'.
Step-by-step solution:
- Step 1: Multiply the first equation by 2 and the second equation by 5 to make the coefficients of 'x' have the same absolute value.
2 * (6z - 5x) = 2 * 2 => 12z - 10x = 4
5 * (4z - 2x) = 5 * 10 => 20z - 10x = 50 - Step 2: Now we have:
12z - 10x = 4
20z - 10x = 50 - Step 3: Subtract the first modified equation from the second modified equation to eliminate 'x'.
(20z - 10x) - (12z - 10x) = 50 - 4
20z - 10x - 12z + 10x = 46
8z = 46 - Step 4: Solve for 'z'.
z = 46 / 8
z = 23 / 4 - Step 5: Substitute the value of 'z' (23/4) back into the original second equation (4z - 2x = 10) to solve for 'x'.
4 * (23/4) - 2x = 10
23 - 2x = 10 - Step 6: Solve for 'x'.
-2x = 10 - 23
-2x = -13
x = -13 / -2
x = 13 / 2
Answer: z = 23/4, x = 13/2
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