Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{6}+\frac{3}{16}=\frac{1}{5}x+5\)
2. \(\frac{x}{2}-\frac{4}{11}=\frac{6}{5}x-2\)
3. \(\frac{x}{2}-\frac{3}{13}=\frac{4}{5}x-3\)
4. \(\frac{x}{5}+\frac{3}{13}=\frac{1}{6}x+3\)
5. \(\frac{x}{7}+\frac{5}{9}=\frac{-7}{5}x-6\)
6. \(\frac{x}{4}+\frac{5}{8}=\frac{-8}{3}x-4\)
7. \(\frac{x}{2}-\frac{2}{9}=\frac{-2}{5}x-1\)
8. \(\frac{x}{2}+\frac{5}{16}=\frac{2}{5}x-1\)
9. \(\frac{x}{2}-\frac{5}{6}=\frac{1}{5}x-4\)
10. \(\frac{x}{6}-\frac{4}{15}=\frac{4}{5}x+8\)
11. \(\frac{x}{3}-\frac{4}{9}=\frac{5}{4}x+2\)
12. \(\frac{x}{2}+\frac{3}{16}=\frac{7}{3}x+3\)

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Reeks met breuken waarbij we ervoor kiezen om de breuken weg te werken. Neem je ZRM!

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x+\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 48x+1200 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-48x} & = & \color{red}{48x} +1200 \color{blue}{-48x} \color{blue}{-45} \\\Leftrightarrow & -8x& = & 1155 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{1155}{-8} \\\Leftrightarrow & x = \frac{-1155}{8} & & \\ & V = \left\{ \frac{-1155}{8} \right\} & \\\end{align}\)
2. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{132}{ \color{blue}{110} }x-\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & 132x-220 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-132x} & = & \color{red}{132x} -220 \color{blue}{-132x} \color{blue}{+40} \\\Leftrightarrow & -77x& = & -180 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-180}{-77} \\\Leftrightarrow & x = \frac{180}{77} & & \\ & V = \left\{ \frac{180}{77} \right\} & \\\end{align}\)
3. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{4}{5}x-3 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{104}{ \color{blue}{130} }x-\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & 104x-390 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{-104x} & = & \color{red}{104x} -390 \color{blue}{-104x} \color{blue}{+30} \\\Leftrightarrow & -39x& = & -360 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-360}{-39} \\\Leftrightarrow & x = \frac{120}{13} & & \\ & V = \left\{ \frac{120}{13} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+90& = & 65x+1170 \\\Leftrightarrow & 78x \color{red}{+90} \color{blue}{-90} \color{blue}{-65x} & = & \color{red}{65x} +1170 \color{blue}{-65x} \color{blue}{-90} \\\Leftrightarrow & 13x& = & 1080 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{1080}{13} \\\Leftrightarrow & x = \frac{1080}{13} & & \\ & V = \left\{ \frac{1080}{13} \right\} & \\\end{align}\)
5. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{315.} (\frac{45x}{ \color{blue}{315} }+ \frac{ 175 }{ \color{blue}{315} })& = & (\frac{-441}{ \color{blue}{315} }x-\frac{1890}{ \color{blue}{315} }) \color{blue}{.315} \\\Leftrightarrow & 45x+175& = & -441x-1890 \\\Leftrightarrow & 45x \color{red}{+175} \color{blue}{-175} \color{blue}{+441x} & = & \color{red}{-441x} -1890 \color{blue}{+441x} \color{blue}{-175} \\\Leftrightarrow & 486x& = & -2065 \\\Leftrightarrow & \frac{486x}{ \color{red}{486} }& = & \frac{-2065}{486} \\\Leftrightarrow & x = \frac{-2065}{486} & & \\ & V = \left\{ \frac{-2065}{486} \right\} & \\\end{align}\)
6. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-64}{ \color{blue}{24} }x-\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+15& = & -64x-96 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{+64x} & = & \color{red}{-64x} -96 \color{blue}{+64x} \color{blue}{-15} \\\Leftrightarrow & 70x& = & -111 \\\Leftrightarrow & \frac{70x}{ \color{red}{70} }& = & \frac{-111}{70} \\\Leftrightarrow & x = \frac{-111}{70} & & \\ & V = \left\{ \frac{-111}{70} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & -36x-90 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{+36x} & = & \color{red}{-36x} -90 \color{blue}{+36x} \color{blue}{+20} \\\Leftrightarrow & 81x& = & -70 \\\Leftrightarrow & \frac{81x}{ \color{red}{81} }& = & \frac{-70}{81} \\\Leftrightarrow & x = \frac{-70}{81} & & \\ & V = \left\{ \frac{-70}{81} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{2}{5}x-1 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{32}{ \color{blue}{80} }x-\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & 32x-80 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{-32x} & = & \color{red}{32x} -80 \color{blue}{-32x} \color{blue}{-25} \\\Leftrightarrow & 8x& = & -105 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-105}{8} \\\Leftrightarrow & x = \frac{-105}{8} & & \\ & V = \left\{ \frac{-105}{8} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-25& = & 6x-120 \\\Leftrightarrow & 15x \color{red}{-25} \color{blue}{+25} \color{blue}{-6x} & = & \color{red}{6x} -120 \color{blue}{-6x} \color{blue}{+25} \\\Leftrightarrow & 9x& = & -95 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{-95}{9} \\\Leftrightarrow & x = \frac{-95}{9} & & \\ & V = \left\{ \frac{-95}{9} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{4}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 24x+240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-24x} & = & \color{red}{24x} +240 \color{blue}{-24x} \color{blue}{+8} \\\Leftrightarrow & -19x& = & 248 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{248}{-19} \\\Leftrightarrow & x = \frac{-248}{19} & & \\ & V = \left\{ \frac{-248}{19} \right\} & \\\end{align}\)
11. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{5}{4}x+2 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{45}{ \color{blue}{36} }x+\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-16& = & 45x+72 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{-45x} & = & \color{red}{45x} +72 \color{blue}{-45x} \color{blue}{+16} \\\Leftrightarrow & -33x& = & 88 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{88}{-33} \\\Leftrightarrow & x = \frac{-8}{3} & & \\ & V = \left\{ \frac{-8}{3} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 112x+144 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-112x} & = & \color{red}{112x} +144 \color{blue}{-112x} \color{blue}{-9} \\\Leftrightarrow & -88x& = & 135 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{135}{-88} \\\Leftrightarrow & x = \frac{-135}{88} & & \\ & V = \left\{ \frac{-135}{88} \right\} & \\\end{align}\)