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

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x-\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & 99x-924 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{-99x} & = & \color{red}{99x} -924 \color{blue}{-99x} \color{blue}{+48} \\\Leftrightarrow & -55x& = & -876 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-876}{-55} \\\Leftrightarrow & x = \frac{876}{55} & & \\ & V = \left\{ \frac{876}{55} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-16& = & 15x+240 \\\Leftrightarrow & 20x \color{red}{-16} \color{blue}{+16} \color{blue}{-15x} & = & \color{red}{15x} +240 \color{blue}{-15x} \color{blue}{+16} \\\Leftrightarrow & 5x& = & 256 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{256}{5} \\\Leftrightarrow & x = \frac{256}{5} & & \\ & V = \left\{ \frac{256}{5} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & -50x+150 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+50x} & = & \color{red}{-50x} +150 \color{blue}{+50x} \color{blue}{-8} \\\Leftrightarrow & 65x& = & 142 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{142}{65} \\\Leftrightarrow & x = \frac{142}{65} & & \\ & V = \left\{ \frac{142}{65} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-18& = & 21x+252 \\\Leftrightarrow & 28x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} +252 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & 7x& = & 270 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{270}{7} \\\Leftrightarrow & x = \frac{270}{7} & & \\ & V = \left\{ \frac{270}{7} \right\} & \\\end{align}\)
5. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-308}{ \color{blue}{220} }x-\frac{1320}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+40& = & -308x-1320 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{+308x} & = & \color{red}{-308x} -1320 \color{blue}{+308x} \color{blue}{-40} \\\Leftrightarrow & 363x& = & -1360 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{-1360}{363} \\\Leftrightarrow & x = \frac{-1360}{363} & & \\ & V = \left\{ \frac{-1360}{363} \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{11}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 42 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+42& = & 77x+1078 \\\Leftrightarrow & 22x \color{red}{+42} \color{blue}{-42} \color{blue}{-77x} & = & \color{red}{77x} +1078 \color{blue}{-77x} \color{blue}{-42} \\\Leftrightarrow & -55x& = & 1036 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{1036}{-55} \\\Leftrightarrow & x = \frac{-1036}{55} & & \\ & V = \left\{ \frac{-1036}{55} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{9}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+50& = & 45x-270 \\\Leftrightarrow & 18x \color{red}{+50} \color{blue}{-50} \color{blue}{-45x} & = & \color{red}{45x} -270 \color{blue}{-45x} \color{blue}{-50} \\\Leftrightarrow & -27x& = & -320 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-320}{-27} \\\Leftrightarrow & x = \frac{320}{27} & & \\ & V = \left\{ \frac{320}{27} \right\} & \\\end{align}\)
8. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{1}{4}x-8 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 21 }{ \color{blue}{112} })& = & (\frac{28}{ \color{blue}{112} }x-\frac{896}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+21& = & 28x-896 \\\Leftrightarrow & 16x \color{red}{+21} \color{blue}{-21} \color{blue}{-28x} & = & \color{red}{28x} -896 \color{blue}{-28x} \color{blue}{-21} \\\Leftrightarrow & -12x& = & -917 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{-917}{-12} \\\Leftrightarrow & x = \frac{917}{12} & & \\ & V = \left\{ \frac{917}{12} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+24& = & 39x-624 \\\Leftrightarrow & 26x \color{red}{+24} \color{blue}{-24} \color{blue}{-39x} & = & \color{red}{39x} -624 \color{blue}{-39x} \color{blue}{-24} \\\Leftrightarrow & -13x& = & -648 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-648}{-13} \\\Leftrightarrow & x = \frac{648}{13} & & \\ & V = \left\{ \frac{648}{13} \right\} & \\\end{align}\)
10. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{7}{4}x+2 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{455}{ \color{blue}{260} }x+\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & 455x+520 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{-455x} & = & \color{red}{455x} +520 \color{blue}{-455x} \color{blue}{+60} \\\Leftrightarrow & -403x& = & 580 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{580}{-403} \\\Leftrightarrow & x = \frac{-580}{403} & & \\ & V = \left\{ \frac{-580}{403} \right\} & \\\end{align}\)
11. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 70 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{154}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+70& = & 77x-154 \\\Leftrightarrow & 22x \color{red}{+70} \color{blue}{-70} \color{blue}{-77x} & = & \color{red}{77x} -154 \color{blue}{-77x} \color{blue}{-70} \\\Leftrightarrow & -55x& = & -224 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-224}{-55} \\\Leftrightarrow & x = \frac{224}{55} & & \\ & V = \left\{ \frac{224}{55} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -140x+504 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+140x} & = & \color{red}{-140x} +504 \color{blue}{+140x} \color{blue}{-48} \\\Leftrightarrow & 161x& = & 456 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{456}{161} \\\Leftrightarrow & x = \frac{456}{161} & & \\ & V = \left\{ \frac{456}{161} \right\} & \\\end{align}\)