Phương trình bậc nhất một ẩn - Hỏi đáp

Chắc chưa học đường chéo vậy làm theo cách thông thường nhé.

Gọi khối lượng của axit 5% là m.

Khối lượng của chất tan trong axit 10% là:

\(m\left(10\%\right)=200\cdot10\%=20\left(g\right)\)

Khối lượng của chất tan trong axit 5% là:

\(m\left(5\%\right)=0,05m\left(g\right)\)

Khối lượng của chất tan trong axit 8% là:

\(m\left(8\%\right)=0,08\left(m+200\right)\left(g\right)\)

Ta có:

\(0,08\left(m+200\right)=20+0,05m\)

\(\Leftrightarrow m=\dfrac{400}{3}\left(g\right)\)

Tui dùng sơ đồ đường chéo giải toán mấy chế ạ :)) lạc đề :V ra kết quả 400/3 g nhé ...

Đặt A=\(\left(\dfrac{a-b}{c}+\dfrac{b-c}{a}+\dfrac{c-a}{b}\right)\left(\dfrac{c}{a-b}+\dfrac{a}{b-c}+\dfrac{b}{c-a}\right)\)

Gọi \(\dfrac{a-b}{c}=x\); \(\dfrac{b-c}{a}=y\); \(\dfrac{c-a}{b}=z\) => \(\dfrac{c}{a-b}=\dfrac{1}{x};\dfrac{a}{b-c}=\dfrac{1}{y};\dfrac{b}{c-a}=\dfrac{1}{z}\)

=> A=(x+y+z)\(\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)

\(=1+\dfrac{x}{y}+\dfrac{x}{z}+\dfrac{y}{x}+1+\dfrac{y}{z}+\dfrac{z}{x}+\dfrac{z}{y}+1\)

= \(\dfrac{x+z}{y}+\dfrac{x+y}{z}+\dfrac{z+y}{x}+3\)

Lại có: \(\dfrac{z+y}{x}=\dfrac{\dfrac{b-c}{a}+\dfrac{c-a}{b}}{\dfrac{a-b}{c}}\) = \(\dfrac{b^2-bc+ac-a^2}{ab}.\dfrac{c}{a-b}\)= \(\dfrac{\left(b-a\right)\left(a+b\right)-c\left(b-a\right)}{ab}.\dfrac{c}{a-b}\) =\(\dfrac{\left(b-a\right)\left(a+b-c\right)}{ab}.\dfrac{c}{a-b}\) = \(\dfrac{-\left(a-b\right)\left(a+b-c\right)}{ab}.\dfrac{c}{a-b}\)= \(\dfrac{\left(-a-b+c\right).c}{ab}\) ⁽¹⁾

Lại có: a+b+c=0 <=> c=-a-b

Thay vào (1) ta được:\(\dfrac{z+y}{x}\)= \(\dfrac{2c^2}{ab}\)

Tương tự ta chứng minh được: \(\dfrac{x+z}{y}=\dfrac{2a^2}{bc}\) ; \(\dfrac{x+y}{z}=\dfrac{2b^2}{ac}\)

=> A=\(\dfrac{2a^2}{bc}+\dfrac{2b^2}{ac}+\dfrac{2c^2}{ab}\)+3 = \(\dfrac{2a^3}{abc}+\dfrac{2b^3}{abc}+\dfrac{2c^3}{abc}+3=\dfrac{2\left(a^3+b^3+c^3\right)}{abc}+3\)

ta chứng minh được \(a^3+b^3+c^3=3abc\) khi a+b+c=0

=> \(A=\dfrac{2.3abc}{abc}+3=6+3=9\)

Vậy A=9

<=> \(\left(\dfrac{a-b}{c}+\dfrac{b-c}{a}+\dfrac{c-a}{b}\right)\left(\dfrac{c}{a-b}+\dfrac{a}{b-c}+\dfrac{b}{c-a}\right)\)=9

Cau 2

Goi x la do dai quang duong AB (x>0)

Thoi gian di tu A-> B la \(\dfrac{x}{40}h\)

Thoi gian di tu B ve A la \(\dfrac{x}{30}h\)

Vi thoi gian ve nhieu hon thoi gian di la 30p =\(\dfrac{1}{2}h\)

nen ta co pt

\(\dfrac{x}{30}-\dfrac{x}{40}=\dfrac{1}{2}\)

\(\Leftrightarrow\)4x -3x=60

\(\Rightarrow\)x = 60 km

Vay quang duong AB dai 60 km

Cau 1

Goi x la do dai quang duong AB (x> 0)

Thoi gian di cua nguoi do tu A-> B la \(\dfrac{x}{25}h\)

Thoi gian ve tu B -> A la \(\dfrac{x}{30}h\)

Vi thoi gian ve it hon thoi gian di la 20 p=\(\dfrac{1}{3}h\) nen ta co phuong trinh

\(\dfrac{x}{25}-\dfrac{x}{30}=\dfrac{1}{3}\)

\(\Leftrightarrow\)\(\dfrac{6x}{150}-\dfrac{5x}{150}=\dfrac{50}{150}\)

\(\Leftrightarrow\)6x - 5x = 50

\(\Rightarrow\)x = 50 km

Một miếng hợp kim đồng và thiết có khối lượng 12kg, trong đó có chứa 45% đồng. Hỏi phải thêm vào đó bao nhiêu kg thiếc nguyên chất để được một hợp kim mới có chứa 40% đồng
Giải
Gọi khối lượng của thiếc thêm vào là x (x\(\in\)Z)(x>0)
=> Theo đề ta có:
- miếng hợp kim óc m=12kg, chứa 45%đồng => m_Cu=45%\(\times12\)(1)
-khi thêm thiếc vào,khối lượng của miếng hợp kim là 12+x mà khối lượng của đồng chỉ chiếm 40%
=> khối lượng của miếng đồng trong hợp kim sau khi thêm thiếc: m_cu=(12+x)\(\times40\%\) (2)

Vì khối lượng của đồng trước và sau không đổi nê từ (1) và (2) ta có phương trình:

45%\(\times12=\left(12+x\right)\times40\%\)
5,4=4,8+40%x
0,6=40%x
=>x=1,5(kg) (TMĐK)
Vậy, khối lượng thiếc thêm vào là 1,5 kg

Đổi 30 phút = \(\frac{1}{2}\)h

Gọi quàng đường AB là x (km)

điều kiện x>0

khi đó:

thời gian xe oto đi từ A đến B là \(\frac{x}{60}\)(h)

thời gian xe oto đi từ B về A là \(\frac{x}{80}\)(h)

Vì thời gian về ít hơn thời gian đi là \(\frac{1}{2}\)h nên ta có phương trình

\(\frac{x}{60}-\frac{x}{80}=\frac{1}{2}\)

\(\Leftrightarrow4x-3x=120\)

\(\Leftrightarrow x=120\)(km)

Vậy quảng đường AB dài 120 km

Đổi 30 phút = 0,5 giờ

Gọi s ( km ) là độ dài quãng đg từ tp A đến tp B ( s > 0 )

+ Thời gian ô tô đi từ tp A đến tp B là :

\(\frac{s}{60}\) ( h )

+ Thời gian ô tô đi từ tp B về tp A là :

\(\frac{s}{80}\) ( h )

+ Ta có pt : \(\frac{s}{60}-\frac{s}{80}=0,5\)

\(\Leftrightarrow\frac{4s-3s}{240}=0,5\) \(\Leftrightarrow s=120\) ( TM )

Vậy độ dài quãng đg từ tp A đến tp B là 120 km.

ta có hpt :\(\left\{{}\begin{matrix}\dfrac{s}{45}-\dfrac{s}{60}=t\\\dfrac{s-30}{45}-\left(\dfrac{\dfrac{S}{2}}{60}+\dfrac{\dfrac{s}{2}-30}{55}\right)=t\end{matrix}\right.\)

t là thời thời gian giữa hai xe xuất phát

tự giải đi

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}}{ }}{ }}{ }}{ }}{ }}{ }}{ }}{ }}{ }}{ }}{ }}{ }\dfrac{ 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}{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{ 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\(A=\frac{5x^2-10x+10-3x^2+6x-3}{x^2-2x+2}=\frac{5\left(x^2-2x+2\right)}{x^2-2x+2}-\frac{3\left(x^2-2x+1\right)}{x^2-2x+2}\)

\(A=5-\frac{3\left(x-1\right)^2}{\left(x-1\right)^2+1}\)

Do \(\left\{{}\begin{matrix}3\left(x-1\right)^2\ge0\\\left(x-1\right)^2+1>0\end{matrix}\right.\) \(\Rightarrow\frac{3\left(x-1\right)^2}{\left(x-1\right)^2+1}\ge0\) \(\forall x\)

\(\Rightarrow A\le5\)

\(\Rightarrow A_{max}=5\) khi \(x=1\)

Ta có: mx + 1 = 3x + m

<=> (m - 3)x = m - 1

Với m \(\ne\) 3 phương trình có nghiệm x = \(\dfrac{m-1}{m-3}=1+\dfrac{2}{m-3}\)

Để x thuộc N thì 2 \(⋮\) (m - 3) = 1 hoặc m - 3 = 2

m - 3 = 1 => m = 4

m - 3 = 2 => m = 5

Vậy m = 4 hoặc m = 5

Nguyễn TrươngNguyễn Việt LâmNguyenTruong Viet TruongKhôi BùiAkai HarumaÁnh LêDƯƠNG PHAN KHÁNH DƯƠNGPhùng Tuệ Minhsaint suppapong udomkaewkanjana

PT: 90%.x+85%.(1200000-x)=1065000

=>0,9.x+0,85.(1200000-x)=1065000

Xong gọi tên r giải

\(45km/h=12,5m/s\)

Gọi chiều dài xe lửa là \(x\) (m)

Theo bài ra ta có pt:

\(\frac{x+9x}{12,5}=120\Rightarrow10x=1500\)

\(\Rightarrow x=150\left(m\right)\)

\(x^3+4x-6=y\left(x^2+3\right)\Rightarrow y=\frac{x^3+4x-6}{x^2+3}=x+\frac{x-6}{x^2+3}\)

Do \(x;y\) nguyên \(\Rightarrow\frac{x-6}{x^2+3}\) nguyên

Nếu lớp 9 đến đoạn này chỉ cần sử dụng miền giá trị, còn lớp 8 thì chịu khó đánh giá:

\(\frac{x-6}{x^2+3}=\frac{-3x^2-9+3x^2+x+3}{x^2+3}=-3+\frac{3x^2+x+3}{x^2+3}>-3\)

\(\frac{x-6}{x^2+3}=\frac{x^2+3-x^2+x-3}{x^2+3}=1-\frac{x^2-x+3}{x^2+3}< 1\)

Vậy \(-3< \frac{x-6}{x^2+3}< 1\Rightarrow\left[{}\begin{matrix}\frac{x-6}{x^2+3}=-2\Rightarrow x=0\\\frac{x-6}{x^2+3}=-1\left(ko-co-x-nguyen\right)\\\frac{x-6}{x^2+3}=0\Rightarrow x=6\end{matrix}\right.\)

Vậy \(\left(x;y\right)=\left(0;-2\right);\left(6;6\right)\)

Trình bày như vậy khó lắm nếu bn ấy chưa tìm hiểu

BĐT

\(\Leftrightarrow\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=9\)( do a,b,c>0)

\(\Leftrightarrow\left(\frac{a}{b}-2+\frac{b}{a}\right)+\left(\frac{b}{c}-2+\frac{c}{b}\right)+\left(\frac{a}{c}-2+\frac{c}{a}\right)\ge0\)

\(\Leftrightarrow\frac{\left(a-b\right)^2}{ab}+\frac{\left(b-c\right)^2}{bc}+\frac{\left(a-c\right)^2}{ac}\ge0\)(đúng)

Áp dụng BĐT Cauchy-Schwarz dạng phân thức cho các số không âm:

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{9}{a+b+c}\)

\(''=''\Leftrightarrow a=b=c\)