Odpowiedź:
[tex]c) \left\{ \begin{array}{ll} 2x + \frac{1}{3}y = 3 \\ - x - y = 1 \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} 2 * 2 + \frac{1}{3} * (-3) = 3 \\ - 2 - (- 3) = 1 \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} 4 - 1 = 3 \\ - 2 + 3 = 1 \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} 3 = 3 \\ 1 = 1 \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} L = R \\ L = R \end{array}[/tex]
Para x = 2, y = -3 spełnia układ równań.
d) [tex]\left\{ \begin{array}{ll} \frac{1}{2}x + \frac{2}{3}y = 2 \\ \frac{5}{2}y - \frac{3}{4}x = 1 \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} \frac{1}{2} * 2 + \frac{2}{3} * (-3) = 2 \\ \frac{5}{2}*(-3) - \frac{3}{4}*2 = 1 \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} 1 - 2 = 2 \\ - 7,5 - 1,5 = 1 \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} - 1 = 2 \\ -9 = 1 \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} L \neq R \\ L \neq R \end{array}[/tex]
Para x = 2, y = -3 nie spełnia układu równań.
e) [tex]\left\{ \begin{array}{ll} \frac{1}{3}x - \frac{1}{2}y = 2\frac{1}{6} \\ \frac{x+1}{3} + \frac{y+1}{4} = \frac{3}{2} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} \frac{1}{3}*2 - \frac{1}{2}*(-3) = 2\frac{1}{6} \\ \frac{2+1}{3} + \frac{-3+1}{4} = \frac{3}{2} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} \frac{2}{3} + \frac{3}{2} = \frac{13}{6} \\ 1 + 1 = \frac{3}{2} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} \frac{4}{6} + \frac{9}{6} = \frac{13}{6} \\ 2 = \frac{3}{2} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} \frac{13}{6} = \frac{13}{6} \\ 2 = \frac{3}{2} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} L = R \\ L \neq R \end{array}[/tex]
Para x = 2, y = -3 nie spełnia układu równań.
f) [tex]\left\{ \begin{array}{ll} \frac{1}{4}x + \frac{1}{6}y = 0 \\\frac{x - 1}{2} + \frac{y - 1}{3} = -\frac{5}{6} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} \frac{1}{4}*2 + \frac{1}{6}*(-3) = 0 \\\frac{2 - 1}{2} + \frac{-3 - 1}{3} = -\frac{5}{6} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} \frac{1}{2} - \frac{1}{2} = 0 \\\frac{1}{2} + \frac{- 4}{3} = -\frac{5}{6} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} \frac{1}{2} - \frac{1}{2} = 0 \\\frac{3}{6} + \frac{- 8}{6} = -\frac{5}{6} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} 0 = 0 \\-\frac{5}{6} = -\frac{5}{6} \end{array}[/tex]
[tex]\left\{ \begin{array}{ll} L = R \\ L = R \end{array}[/tex]
Para x = 2, y = -3 spełnia układu równań.