Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{4}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{56}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & 56x+70 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-56x} & = & \color{red}{56x} +70 \color{blue}{-56x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & 30 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{30}{-21} \\\Leftrightarrow & x = \frac{-10}{7} & & \\ & V = \left\{ \frac{-10}{7} \right\} & \\\end{align}\)
2. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{448}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 56x-448 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-56x} & = & \color{red}{56x} -448 \color{blue}{-56x} \color{blue}{-35} \\\Leftrightarrow & -40x& = & -483 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-483}{-40} \\\Leftrightarrow & x = \frac{483}{40} & & \\ & V = \left\{ \frac{483}{40} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{5}{3}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 140x-420 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-140x} & = & \color{red}{140x} -420 \color{blue}{-140x} \color{blue}{+48} \\\Leftrightarrow & -119x& = & -372 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-372}{-119} \\\Leftrightarrow & x = \frac{372}{119} & & \\ & V = \left\{ \frac{372}{119} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{3}{4}x+8 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{117}{ \color{blue}{156} }x+\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & 117x+1248 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{-117x} & = & \color{red}{117x} +1248 \color{blue}{-117x} \color{blue}{+24} \\\Leftrightarrow & -65x& = & 1272 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1272}{-65} \\\Leftrightarrow & x = \frac{-1272}{65} & & \\ & V = \left\{ \frac{-1272}{65} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 33x+528 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-33x} & = & \color{red}{33x} +528 \color{blue}{-33x} \color{blue}{-30} \\\Leftrightarrow & -11x& = & 498 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{498}{-11} \\\Leftrightarrow & x = \frac{-498}{11} & & \\ & V = \left\{ \frac{-498}{11} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 50 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+50& = & -56x-490 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{+56x} & = & \color{red}{-56x} -490 \color{blue}{+56x} \color{blue}{-50} \\\Leftrightarrow & 91x& = & -540 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-540}{91} \\\Leftrightarrow & x = \frac{-540}{91} & & \\ & V = \left\{ \frac{-540}{91} \right\} & \\\end{align}\)
7. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 10 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x-\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+10& = & -49x-84 \\\Leftrightarrow & 4x \color{red}{+10} \color{blue}{-10} \color{blue}{+49x} & = & \color{red}{-49x} -84 \color{blue}{+49x} \color{blue}{-10} \\\Leftrightarrow & 53x& = & -94 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{-94}{53} \\\Leftrightarrow & x = \frac{-94}{53} & & \\ & V = \left\{ \frac{-94}{53} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 7, 6 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 175 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-175& = & -84x+1050 \\\Leftrightarrow & 30x \color{red}{-175} \color{blue}{+175} \color{blue}{+84x} & = & \color{red}{-84x} +1050 \color{blue}{+84x} \color{blue}{+175} \\\Leftrightarrow & 114x& = & 1225 \\\Leftrightarrow & \frac{114x}{ \color{red}{114} }& = & \frac{1225}{114} \\\Leftrightarrow & x = \frac{1225}{114} & & \\ & V = \left\{ \frac{1225}{114} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -42x-150 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} -150 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 47x& = & -142 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-142}{47} \\\Leftrightarrow & x = \frac{-142}{47} & & \\ & V = \left\{ \frac{-142}{47} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-40& = & 65x-650 \\\Leftrightarrow & 26x \color{red}{-40} \color{blue}{+40} \color{blue}{-65x} & = & \color{red}{65x} -650 \color{blue}{-65x} \color{blue}{+40} \\\Leftrightarrow & -39x& = & -610 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-610}{-39} \\\Leftrightarrow & x = \frac{610}{39} & & \\ & V = \left\{ \frac{610}{39} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & -220x+792 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{+220x} & = & \color{red}{-220x} +792 \color{blue}{+220x} \color{blue}{-60} \\\Leftrightarrow & 253x& = & 732 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{732}{253} \\\Leftrightarrow & x = \frac{732}{253} & & \\ & V = \left\{ \frac{732}{253} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & 14x+210 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{-14x} & = & \color{red}{14x} +210 \color{blue}{-14x} \color{blue}{+20} \\\Leftrightarrow & 21x& = & 230 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{230}{21} \\\Leftrightarrow & x = \frac{230}{21} & & \\ & V = \left\{ \frac{230}{21} \right\} & \\\end{align}\)