Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & 10x+30 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{-10x} & = & \color{red}{10x} +30 \color{blue}{-10x} \color{blue}{-9} \\\Leftrightarrow & 5x& = & 21 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{21}{5} \\\Leftrightarrow & x = \frac{21}{5} & & \\ & V = \left\{ \frac{21}{5} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & 12x+108 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{-12x} & = & \color{red}{12x} +108 \color{blue}{-12x} \color{blue}{-16} \\\Leftrightarrow & -3x& = & 92 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{92}{-3} \\\Leftrightarrow & x = \frac{-92}{3} & & \\ & V = \left\{ \frac{-92}{3} \right\} & \\\end{align}\)
3. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+50& = & 65x-910 \\\Leftrightarrow & 26x \color{red}{+50} \color{blue}{-50} \color{blue}{-65x} & = & \color{red}{65x} -910 \color{blue}{-65x} \color{blue}{-50} \\\Leftrightarrow & -39x& = & -960 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-960}{-39} \\\Leftrightarrow & x = \frac{320}{13} & & \\ & V = \left\{ \frac{320}{13} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{135}{ \color{blue}{90} }x+\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & 135x+90 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{-135x} & = & \color{red}{135x} +90 \color{blue}{-135x} \color{blue}{-20} \\\Leftrightarrow & -117x& = & 70 \\\Leftrightarrow & \frac{-117x}{ \color{red}{-117} }& = & \frac{70}{-117} \\\Leftrightarrow & x = \frac{-70}{117} & & \\ & V = \left\{ \frac{-70}{117} \right\} & \\\end{align}\)
5. \(\text{280 is het kleinste gemene veelvoud van 7, 8 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{8}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{280.} (\frac{40x}{ \color{blue}{280} }- \frac{ 105 }{ \color{blue}{280} })& = & (\frac{-112}{ \color{blue}{280} }x-\frac{2240}{ \color{blue}{280} }) \color{blue}{.280} \\\Leftrightarrow & 40x-105& = & -112x-2240 \\\Leftrightarrow & 40x \color{red}{-105} \color{blue}{+105} \color{blue}{+112x} & = & \color{red}{-112x} -2240 \color{blue}{+112x} \color{blue}{+105} \\\Leftrightarrow & 152x& = & -2135 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-2135}{152} \\\Leftrightarrow & x = \frac{-2135}{152} & & \\ & V = \left\{ \frac{-2135}{152} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{50}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 50x+210 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-50x} & = & \color{red}{50x} +210 \color{blue}{-50x} \color{blue}{+4} \\\Leftrightarrow & -35x& = & 214 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{214}{-35} \\\Leftrightarrow & x = \frac{-214}{35} & & \\ & V = \left\{ \frac{-214}{35} \right\} & \\\end{align}\)
7. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{4}{ \color{blue}{20} }x-\frac{20}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & 4x-20 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{-4x} & = & \color{red}{4x} -20 \color{blue}{-4x} \color{blue}{-6} \\\Leftrightarrow & x& = & -26 \\ & V = \left\{ -26 \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -42x-60 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} -60 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 47x& = & -52 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-52}{47} \\\Leftrightarrow & x = \frac{-52}{47} & & \\ & V = \left\{ \frac{-52}{47} \right\} & \\\end{align}\)
9. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & -336x-480 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{+336x} & = & \color{red}{-336x} -480 \color{blue}{+336x} \color{blue}{+45} \\\Leftrightarrow & 376x& = & -435 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{-435}{376} \\\Leftrightarrow & x = \frac{-435}{376} & & \\ & V = \left\{ \frac{-435}{376} \right\} & \\\end{align}\)
10. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 55x+880 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} +880 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & 920 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{920}{-33} \\\Leftrightarrow & x = \frac{-920}{33} & & \\ & V = \left\{ \frac{-920}{33} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 42x+210 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-42x} & = & \color{red}{42x} +210 \color{blue}{-42x} \color{blue}{+90} \\\Leftrightarrow & -7x& = & 300 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{300}{-7} \\\Leftrightarrow & x = \frac{-300}{7} & & \\ & V = \left\{ \frac{-300}{7} \right\} & \\\end{align}\)
12. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{20}{ \color{blue}{120} }x-\frac{840}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-75& = & 20x-840 \\\Leftrightarrow & 24x \color{red}{-75} \color{blue}{+75} \color{blue}{-20x} & = & \color{red}{20x} -840 \color{blue}{-20x} \color{blue}{+75} \\\Leftrightarrow & 4x& = & -765 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-765}{4} \\\Leftrightarrow & x = \frac{-765}{4} & & \\ & V = \left\{ \frac{-765}{4} \right\} & \\\end{align}\)