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Giải bài 6.22 trang 10 sách bài tập toán 8 - Kết nối tri thức với cuộc sống

Cho biểu thức \(P = \frac{x}{{y - 2}} + \frac{{2x - 3y}}{{x - 6}}.\)

ℱTổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

Cho biểu thức \(P = \frac{x}{{y - 2}} + \frac{{2x - 3y}}{{x - 6}}.\) Chứng minh rằng khi x, y thay đổi luôn thỏa mãn điều kiện \(3y - x = 6\) thì P có giá trị không đổi.

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức rút gọn phân thức để chứng minh: + Rút gọn phân thức là biến đổi phân thức đó thành một biểu thức mới bằng nó nhưng đơn giản hơn + Muốn rút gọn một phân thức đại số ta làm như sau: - Phân tích tử và mẫu thành nhân tử (nếu cần) để tìm nhân tử chung; - Chia cả tử và mẫu cho nhân tử chung đó.

Lời giải chi tiết

Vì \(3y - x = 6\) nên \(x = 3y - 6\), thay vào P ta có:\(P = \frac{x}{{y - 2}} + \frac{{2x - 3y}}{{x - 6}} \\= \frac{{3y - 6}}{{y - 2}} + \frac{{2\left( {3y - 6} \right) - 3y}}{{3y - 6 - 6}} \\= \frac{{3\left( {y - 2} \right)}}{{y - 2}} + \frac{{6y - 12 - 3y}}{{3y - 12}}\)\( = 3 + \frac{{3y - 12}}{{3y - 12}} = 3 + 1 = 4\)Vậy khi x, y thay đổi luôn thỏa mãn điều kiện \(3y - x = 6\) thì P có giá trị không đổi.

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