Đáp án C

Ta có  

$$\begin{gathered} \left\{\begin{array}{l}5\left(x+2y\right)-3\left(x-y\right)=99\\ x-3y=7x-4y-17\end{array}\right. \\ \iff \left\{\begin{array}{l}5x+10y-3x+3y=-17\\ x-3y-7x+4y=-17\end{array}\right. \\ \iff \left\{\begin{array}{l}6x+39y=297\\ -6x+y=-17\end{array}\right.\iff \left\{\begin{array}{l}-6x+y=-17\\ 40y=280\end{array}\right. \\ \iff \left\{\begin{array}{l}2x+13y=99\\ -6x+y=-17\end{array}\right.\iff \left\{\begin{array}{l}y=7\\ x=4\end{array}\right. \end{gathered}$$

Vậy hệ phương trình có nghiệm duy nhất (x; y) = (4; 7)

Đáp án A

Thay x = 3; y = −4 vào hệ phương trình ta được

$$\begin{gathered} \left\{\begin{array}{l}2a.3+b\left(-4\right)=-1\\ b.3-a.\left(-4\right)=5\end{array}\right.\iff \left\{\begin{array}{l}6a-4b=-1\\ 4a+3b=5\end{array}\right. \\ \iff \left\{\begin{array}{l}12a-8b=-2\\ 12a+9b=15\end{array}\right.\iff \left\{\begin{array}{l}17b=17\\ 4a+3b=5\end{array}\right. \\ \iff \left\{\begin{array}{l}b=1\\ a=\frac{1}{2}\end{array}\right. \end{gathered}$$

Vậy $a=\frac{1}{2} ; b=1$

Đáp án D

Thay x = 2; y = −3 vào hệ phương trình ta được:

$$\begin{gathered} \left\{\begin{array}{l}4a.2+2b.\left(-3\right)=-3\\ 3b.2+a\left(-3\right)=8\end{array}\right.\iff \left\{\begin{array}{l}8a-6b=-3\\ -3a+6b=8\end{array}\right. \\ \iff \left\{\begin{array}{l}5a=5\\ -3a+6b=8\end{array}\right.\iff \left\{\begin{array}{l}a=1\\ -3.1+6b=8\end{array}\right. \\ \iff \left\{\begin{array}{l}a=1\\ 6b=11\end{array}\right.\iff \left\{\begin{array}{l}a=1\\ b=\frac{11}{6}\end{array}\right. \end{gathered}$$

Vậy $a=1; b=\frac{11}{6}$

Đáp án A

Ta có

$$\begin{gathered} \left\{\begin{array}{l}\frac{2x+1}{3}-\frac{y+1}{4}=\frac{4x-2y+2}{5}\\ \frac{2x-3}{4}-\frac{y-4}{3}=-2x+2y-2\end{array}\right. \\ \iff \left\{\begin{array}{l}40x+20-15y-15=48x-24y+24\\ 6x-9-4y+16=-24x+24y-24\end{array}\right. \\ \iff \left\{\begin{array}{l}8x-9y=-19\\ 30x-28y=-31\end{array}\right.\iff \left\{\begin{array}{l}120x-135=-285\\ 120x-112=-124\end{array}\right. \\ \iff \left\{\begin{array}{l}x=\frac{11}{2}\\ y=7\end{array}\right. \end{gathered}$$

Thay $x=\frac{11}{2}$ ; y = 7 vào phương trình 6mx – 5y = 2m – 66 ta được:

$$\begin{gathered} 6m.\frac{11}{2}-5.7=2m–66 \\ \iff 31m=-31\iff m=-1 \end{gathered}$$

Đáp án C

Ta có

$$\begin{gathered} \left\{\begin{array}{l}\frac{x+1}{4}-\frac{y}{2}=x+y+1\\ \frac{x-2}{2}+\frac{y-1}{3}=x+y-1\end{array}\right. \\ \iff \left\{\begin{array}{l}x+1-2y=4x+4y+4\\ 3x-6+2y-2=6x+6y-6\end{array}\right. \\ \iff \left\{\begin{array}{l}3x+6y=-3\\ 3x+4y=-2\end{array}\right.\iff \left\{\begin{array}{l}y=\frac{-1}{2}\\ x=0\end{array}\right. \end{gathered}$$

Thay x = 0; $y=-\frac{1}{2}$ vào phương trình (m + 2)x + 7my = m – 225 ta được:

$$\begin{gathered} (m+2).0+7m\left(-\frac{1}{2}\right)=m–225 \\ \iff \frac{9}{2}m=225\iff m=50 \end{gathered}$$

Đáp án B

Đường thẳng y = ax + b đi qua điểm A (−4; −2) $\iff$ −4a + b = −2   (1)

Đường thẳng y = ax + b đi qua điểm B (2; 1) $\iff$ 2a + b = 1            (2)

Từ (1) và (2) ta có hệ:

$\begin{array}{l}\left\{\begin{array}{l}-4a+b=-2\\ 2a+b=1\end{array}\right.\iff \left\{\begin{array}{l}-6a=-3\\ 2a+b=1\end{array}\right.\\ \iff \left\{\begin{array}{l}a=\frac{1}{2}\\ 2.\frac{1}{2}+b=1\end{array}\right.\iff \left\{\begin{array}{l}a=\frac{1}{2}\\ b=0\end{array}\right.\end{array}$

Vậy $a = \frac{1}{2}; b = 0$

Đáp án A

Đường thẳng y = ax + b đi qua điểm A ($\sqrt{3}$ ; 2) $\iff$ − a + 2b = 2           (1)

Đường thẳng y = ax + b đi qua điểm B (0; 2) $\iff$ 0.a + b = 2           (2)

Từ (1) và (2) ta có hệ:

$\left\{\begin{array}{l}\sqrt{3}a+b=2\\ 0.a+b=2\end{array}\right.\iff \left\{\begin{array}{l}b=2\\ \sqrt{3}a+2=2\end{array}\right.\iff \left\{\begin{array}{l}a=0\\ b=2\end{array}\right.$

Vậy a = 0; b = 2

Đáp án D

Ta có  

$$\begin{gathered} \left\{\begin{array}{l}2\left(x+y\right)-3\left(x-y\right)=4\\ x+4y=2x-y+5\end{array}\right.\iff \left\{\begin{array}{l}2x+2y-3x+3y=4\\ x+4y-2x+y=5\end{array}\right. \\ \iff \left\{\begin{array}{l}-x+5y=4\\ -x+5y=5\end{array}\right.\iff \left\{\begin{array}{l}0=1\\ -x+5y=5\end{array}\right.\left(VL\right) \end{gathered}$$

Vậy hệ phương trình vô nghiệm

Đáp án C

ĐK: x $\ge$ 1; y $\ge$ 1

Ta có  

$$\begin{gathered} \left\{\begin{array}{l}3\sqrt{x-1}+2\sqrt{y}=13\\ 2\sqrt{x-1}-\sqrt{y}=4\end{array}\right.\iff \left\{\begin{array}{l}3\sqrt{x-1}+2\sqrt{y}=13\\ 4\sqrt{x-1}-2\sqrt{y}=8\end{array}\right. \\ \iff \left\{\begin{array}{l}2\sqrt{x-1}-\sqrt{y}=4\\ 7\sqrt{x-1}=21\end{array}\right.\begin{array}{l}\iff \left\{\begin{array}{l}\sqrt{x-1}=3\\ 3.3+2\sqrt{y}=13\end{array}\right.\\ \end{array} \\ \iff \left\{\begin{array}{l}x-1=9\\ 2\sqrt{y}=4\end{array}\right.\iff \left\{\begin{array}{l}x=10\\ y=4\end{array}\right.(thỏa mãn) \end{gathered}$$

Vậy hệ phương trình có nghiệm duy nhất (x; y) = (10; 4)

Nên x – y = 10 – 4 = 6

Đáp án A

Ta có

$$\begin{gathered} \left\{\begin{array}{l}x+\frac{y}{2}=\frac{2x-3}{2}\\ \frac{x}{2}+3y=\frac{25-9y}{8}\end{array}\right.\iff \left\{\begin{array}{l}2x+y=2x-3\\ 4x+24y=25-9y\end{array}\right. \\ \iff \left\{\begin{array}{l}y=-3\\ 4x+33y=25\end{array}\right.\iff \left\{\begin{array}{l}x=31\\ y=-3\end{array}\right. \end{gathered}$$

Vậy hệ phương trình có nghiệm duy nhất (x; y) = (31; −3)

$\Rightarrow$ x > 0; y < 0

Đáp án D

Ta có

$$\begin{gathered} \left\{\begin{array}{l}\frac{x+y}{5}=\frac{x-y}{3}\\ \frac{x}{4}=\frac{y}{2}+1\end{array}\right.\iff \left\{\begin{array}{l}3x+3y=5x-5y\\ x=2y+4\end{array}\right. \\ \iff \left\{\begin{array}{l}2x=8y\\ x=2y+4\end{array}\right.\iff \left\{\begin{array}{l}x=4y\\ x=2y+4\end{array}\right. \\ \iff \left\{\begin{array}{l}x=4y\\ 2y-4=0\end{array}\right.\iff \left\{\begin{array}{l}y=2\\ x=8\end{array}\right. \end{gathered}$$

Vậy hệ phương trình có nghiệm duy nhất (x; y) = (2; 8)

Đáp án B

ĐK: x $\ge$ −3; y $\ge$ −1

Ta có:

$$\begin{gathered} \left\{\begin{array}{l}\sqrt{x+3}-2\sqrt{y+1}=2\\ 2\sqrt{x+3}+\sqrt{y+1}=4\end{array}\right.\iff \left\{\begin{array}{l}2\sqrt{x+3}-4\sqrt{y+1}=4\\ 2\sqrt{x+3}+\sqrt{y+1}=4\end{array}\right. \\ \iff \left\{\begin{array}{l}\sqrt{x+3}-2\sqrt{y+1}=2\\ -5\sqrt{y+1}=0\end{array}\right.\iff \left\{\begin{array}{l}y=-1\\ \sqrt{x+3}-2.\sqrt{\left(-1\right)+1}=2\end{array}\right. \\ \iff \left\{\begin{array}{l}y=-1\\ \sqrt{x+3}=2\end{array}\right.\iff \left\{\begin{array}{l}y=-1\\ x+3=4\end{array}\right.\iff \left\{\begin{array}{l}y=-1\\ x=1\end{array}\right.\left(tm\right) \end{gathered}$$

Vậy hệ phương trình có nghiệm duy nhất (x; y) = (1; −1)

Nên x + y = 1 + (−1) = 0

Đáp án B

Ta có

$$\begin{gathered} \left\{\begin{array}{l}\left(x-3\right)\left(2y+5\right)=\left(2x+7\right)\left(y-1\right)\\ \left(4x+1\right)\left(3y-6\right)=\left(6x-1\right)\left(2y+3\right)\end{array}\right. \\ \iff \left\{\begin{array}{l}7x-13y=8\\ -42x+5y=3\end{array}\right.\iff \left\{\begin{array}{l}42x-78y=48\\ -42x+5y=3\end{array}\right. \end{gathered}$$

Đáp án D

Ta có:

$$\begin{gathered} \left\{\begin{array}{l}\left(2x+4\right)\left(6-y\right)=\left(11-x\right)\left(2y+6\right)\\ 3\left(x+1\right)\left(y+1\right)=\left(3x+4\right)\left(y+2\right)\end{array}\right. \\ \iff \left\{\begin{array}{l}12x-2xy+24-4y=22y+66-2xy-6x\\ 3xy+3x+3y+3=3xy+6x+4y+8\end{array}\right. \\ \iff \left\{\begin{array}{l}18x-26y-42=0\\ -3x-y-5=0\end{array}\right.\iff \left\{\begin{array}{l}9x-13y=21\\ 3x+y=-5\end{array}\right. \end{gathered}$$