Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}+\frac{4}{11}=\frac{1}{6}x+5\)
2. \(\frac{x}{4}+\frac{3}{10}=\frac{1}{3}x+1\)
3. \(\frac{x}{7}-\frac{5}{14}=\frac{1}{4}x+7\)
4. \(\frac{x}{2}-\frac{2}{13}=\frac{7}{3}x+7\)
5. \(\frac{x}{3}+\frac{4}{13}=\frac{6}{5}x-8\)
6. \(\frac{x}{7}+\frac{3}{10}=\frac{-4}{5}x-5\)
7. \(\frac{x}{6}+\frac{5}{16}=\frac{-4}{5}x-5\)
8. \(\frac{x}{6}+\frac{2}{13}=\frac{7}{5}x+7\)
9. \(\frac{x}{5}+\frac{2}{15}=\frac{5}{4}x-2\)
10. \(\frac{x}{4}-\frac{4}{15}=\frac{4}{3}x-7\)
11. \(\frac{x}{2}+\frac{5}{14}=\frac{7}{3}x-1\)
12. \(\frac{x}{2}+\frac{5}{8}=\frac{-2}{3}x+2\)

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x+\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & 55x+1650 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{-55x} & = & \color{red}{55x} +1650 \color{blue}{-55x} \color{blue}{-120} \\\Leftrightarrow & 11x& = & 1530 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{1530}{11} \\\Leftrightarrow & x = \frac{1530}{11} & & \\ & V = \left\{ \frac{1530}{11} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & 20x+60 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{-20x} & = & \color{red}{20x} +60 \color{blue}{-20x} \color{blue}{-18} \\\Leftrightarrow & -5x& = & 42 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{42}{-5} \\\Leftrightarrow & x = \frac{-42}{5} & & \\ & V = \left\{ \frac{-42}{5} \right\} & \\\end{align}\)
3. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{14}& = & \frac{1}{4}x+7 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 10 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x+\frac{196}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-10& = & 7x+196 \\\Leftrightarrow & 4x \color{red}{-10} \color{blue}{+10} \color{blue}{-7x} & = & \color{red}{7x} +196 \color{blue}{-7x} \color{blue}{+10} \\\Leftrightarrow & -3x& = & 206 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{206}{-3} \\\Leftrightarrow & x = \frac{-206}{3} & & \\ & V = \left\{ \frac{-206}{3} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{182}{ \color{blue}{78} }x+\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & 182x+546 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{-182x} & = & \color{red}{182x} +546 \color{blue}{-182x} \color{blue}{+12} \\\Leftrightarrow & -143x& = & 558 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{558}{-143} \\\Leftrightarrow & x = \frac{-558}{143} & & \\ & V = \left\{ \frac{-558}{143} \right\} & \\\end{align}\)
5. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{234}{ \color{blue}{195} }x-\frac{1560}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+60& = & 234x-1560 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-234x} & = & \color{red}{234x} -1560 \color{blue}{-234x} \color{blue}{-60} \\\Leftrightarrow & -169x& = & -1620 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-1620}{-169} \\\Leftrightarrow & x = \frac{1620}{169} & & \\ & V = \left\{ \frac{1620}{169} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 21 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+21& = & -56x-350 \\\Leftrightarrow & 10x \color{red}{+21} \color{blue}{-21} \color{blue}{+56x} & = & \color{red}{-56x} -350 \color{blue}{+56x} \color{blue}{-21} \\\Leftrightarrow & 66x& = & -371 \\\Leftrightarrow & \frac{66x}{ \color{red}{66} }& = & \frac{-371}{66} \\\Leftrightarrow & x = \frac{-371}{66} & & \\ & V = \left\{ \frac{-371}{66} \right\} & \\\end{align}\)
7. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -192x-1200 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+192x} & = & \color{red}{-192x} -1200 \color{blue}{+192x} \color{blue}{-75} \\\Leftrightarrow & 232x& = & -1275 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{-1275}{232} \\\Leftrightarrow & x = \frac{-1275}{232} & & \\ & V = \left\{ \frac{-1275}{232} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{546}{ \color{blue}{390} }x+\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 546x+2730 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-546x} & = & \color{red}{546x} +2730 \color{blue}{-546x} \color{blue}{-60} \\\Leftrightarrow & -481x& = & 2670 \\\Leftrightarrow & \frac{-481x}{ \color{red}{-481} }& = & \frac{2670}{-481} \\\Leftrightarrow & x = \frac{-2670}{481} & & \\ & V = \left\{ \frac{-2670}{481} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{5}{4}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+8& = & 75x-120 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-75x} & = & \color{red}{75x} -120 \color{blue}{-75x} \color{blue}{-8} \\\Leftrightarrow & -63x& = & -128 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{-128}{-63} \\\Leftrightarrow & x = \frac{128}{63} & & \\ & V = \left\{ \frac{128}{63} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 80x-420 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-80x} & = & \color{red}{80x} -420 \color{blue}{-80x} \color{blue}{+16} \\\Leftrightarrow & -65x& = & -404 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-404}{-65} \\\Leftrightarrow & x = \frac{404}{65} & & \\ & V = \left\{ \frac{404}{65} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{14}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 15 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+15& = & 98x-42 \\\Leftrightarrow & 21x \color{red}{+15} \color{blue}{-15} \color{blue}{-98x} & = & \color{red}{98x} -42 \color{blue}{-98x} \color{blue}{-15} \\\Leftrightarrow & -77x& = & -57 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-57}{-77} \\\Leftrightarrow & x = \frac{57}{77} & & \\ & V = \left\{ \frac{57}{77} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x+\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+15& = & -16x+48 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+16x} & = & \color{red}{-16x} +48 \color{blue}{+16x} \color{blue}{-15} \\\Leftrightarrow & 28x& = & 33 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{33}{28} \\\Leftrightarrow & x = \frac{33}{28} & & \\ & V = \left\{ \frac{33}{28} \right\} & \\\end{align}\)