Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{462}{ \color{blue}{330} }x+\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & 462x+2310 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-462x} & = & \color{red}{462x} +2310 \color{blue}{-462x} \color{blue}{-60} \\\Leftrightarrow & -407x& = & 2250 \\\Leftrightarrow & \frac{-407x}{ \color{red}{-407} }& = & \frac{2250}{-407} \\\Leftrightarrow & x = \frac{-2250}{407} & & \\ & V = \left\{ \frac{-2250}{407} \right\} & \\\end{align}\)
2. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{1}{4}x-8 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 21 }{ \color{blue}{112} })& = & (\frac{28}{ \color{blue}{112} }x-\frac{896}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-21& = & 28x-896 \\\Leftrightarrow & 16x \color{red}{-21} \color{blue}{+21} \color{blue}{-28x} & = & \color{red}{28x} -896 \color{blue}{-28x} \color{blue}{+21} \\\Leftrightarrow & -12x& = & -875 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{-875}{-12} \\\Leftrightarrow & x = \frac{875}{12} & & \\ & V = \left\{ \frac{875}{12} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & -52x-546 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+52x} & = & \color{red}{-52x} -546 \color{blue}{+52x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -570 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-570}{91} \\\Leftrightarrow & x = \frac{-570}{91} & & \\ & V = \left\{ \frac{-570}{91} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 39x-78 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-39x} & = & \color{red}{39x} -78 \color{blue}{-39x} \color{blue}{+12} \\\Leftrightarrow & -13x& = & -66 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-66}{-13} \\\Leftrightarrow & x = \frac{66}{13} & & \\ & V = \left\{ \frac{66}{13} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x-\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & -112x-320 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{+112x} & = & \color{red}{-112x} -320 \color{blue}{+112x} \color{blue}{-15} \\\Leftrightarrow & 132x& = & -335 \\\Leftrightarrow & \frac{132x}{ \color{red}{132} }& = & \frac{-335}{132} \\\Leftrightarrow & x = \frac{-335}{132} & & \\ & V = \left\{ \frac{-335}{132} \right\} & \\\end{align}\)
6. \(\text{20 is het kleinste gemene veelvoud van 5, 10 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{-3}{4}x+5 \\\Leftrightarrow & \color{blue}{20.} (\frac{4x}{ \color{blue}{20} }- \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-15}{ \color{blue}{20} }x+\frac{100}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 4x-6& = & -15x+100 \\\Leftrightarrow & 4x \color{red}{-6} \color{blue}{+6} \color{blue}{+15x} & = & \color{red}{-15x} +100 \color{blue}{+15x} \color{blue}{+6} \\\Leftrightarrow & 19x& = & 106 \\\Leftrightarrow & \frac{19x}{ \color{red}{19} }& = & \frac{106}{19} \\\Leftrightarrow & x = \frac{106}{19} & & \\ & V = \left\{ \frac{106}{19} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{4}{3}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 63 }{ \color{blue}{210} })& = & (\frac{280}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+63& = & 280x+630 \\\Leftrightarrow & 30x \color{red}{+63} \color{blue}{-63} \color{blue}{-280x} & = & \color{red}{280x} +630 \color{blue}{-280x} \color{blue}{-63} \\\Leftrightarrow & -250x& = & 567 \\\Leftrightarrow & \frac{-250x}{ \color{red}{-250} }& = & \frac{567}{-250} \\\Leftrightarrow & x = \frac{-567}{250} & & \\ & V = \left\{ \frac{-567}{250} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 196x-672 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-196x} & = & \color{red}{196x} -672 \color{blue}{-196x} \color{blue}{+36} \\\Leftrightarrow & -175x& = & -636 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-636}{-175} \\\Leftrightarrow & x = \frac{636}{175} & & \\ & V = \left\{ \frac{636}{175} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-45& = & 35x-1260 \\\Leftrightarrow & 42x \color{red}{-45} \color{blue}{+45} \color{blue}{-35x} & = & \color{red}{35x} -1260 \color{blue}{-35x} \color{blue}{+45} \\\Leftrightarrow & 7x& = & -1215 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1215}{7} \\\Leftrightarrow & x = \frac{-1215}{7} & & \\ & V = \left\{ \frac{-1215}{7} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x-336 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} -336 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & -348 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-348}{-7} \\\Leftrightarrow & x = \frac{348}{7} & & \\ & V = \left\{ \frac{348}{7} \right\} & \\\end{align}\)
11. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{-2}{5}x+7 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }- \frac{ 2 }{ \color{blue}{15} })& = & (\frac{-6}{ \color{blue}{15} }x+\frac{105}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x-2& = & -6x+105 \\\Leftrightarrow & 5x \color{red}{-2} \color{blue}{+2} \color{blue}{+6x} & = & \color{red}{-6x} +105 \color{blue}{+6x} \color{blue}{+2} \\\Leftrightarrow & 11x& = & 107 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{107}{11} \\\Leftrightarrow & x = \frac{107}{11} & & \\ & V = \left\{ \frac{107}{11} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & 44x-792 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-44x} & = & \color{red}{44x} -792 \color{blue}{-44x} \color{blue}{-60} \\\Leftrightarrow & -11x& = & -852 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-852}{-11} \\\Leftrightarrow & x = \frac{852}{11} & & \\ & V = \left\{ \frac{852}{11} \right\} & \\\end{align}\)