Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-40& = & 65x-910 \\\Leftrightarrow & 26x \color{red}{-40} \color{blue}{+40} \color{blue}{-65x} & = & \color{red}{65x} -910 \color{blue}{-65x} \color{blue}{+40} \\\Leftrightarrow & -39x& = & -870 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-870}{-39} \\\Leftrightarrow & x = \frac{290}{13} & & \\ & V = \left\{ \frac{290}{13} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -294x+840 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+294x} & = & \color{red}{-294x} +840 \color{blue}{+294x} \color{blue}{-120} \\\Leftrightarrow & 329x& = & 720 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{720}{329} \\\Leftrightarrow & x = \frac{720}{329} & & \\ & V = \left\{ \frac{720}{329} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 126x+1470 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-126x} & = & \color{red}{126x} +1470 \color{blue}{-126x} \color{blue}{+60} \\\Leftrightarrow & -91x& = & 1530 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{1530}{-91} \\\Leftrightarrow & x = \frac{-1530}{91} & & \\ & V = \left\{ \frac{-1530}{91} \right\} & \\\end{align}\)
4. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-18}{ \color{blue}{24} }x+\frac{24}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & -18x+24 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{+18x} & = & \color{red}{-18x} +24 \color{blue}{+18x} \color{blue}{+15} \\\Leftrightarrow & 26x& = & 39 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{39}{26} \\\Leftrightarrow & x = \frac{3}{2} & & \\ & V = \left\{ \frac{3}{2} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{14}& = & \frac{4}{5}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{56}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+25& = & 56x+560 \\\Leftrightarrow & 35x \color{red}{+25} \color{blue}{-25} \color{blue}{-56x} & = & \color{red}{56x} +560 \color{blue}{-56x} \color{blue}{-25} \\\Leftrightarrow & -21x& = & 535 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{535}{-21} \\\Leftrightarrow & x = \frac{-535}{21} & & \\ & V = \left\{ \frac{-535}{21} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+40& = & 55x-660 \\\Leftrightarrow & 22x \color{red}{+40} \color{blue}{-40} \color{blue}{-55x} & = & \color{red}{55x} -660 \color{blue}{-55x} \color{blue}{-40} \\\Leftrightarrow & -33x& = & -700 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-700}{-33} \\\Leftrightarrow & x = \frac{700}{33} & & \\ & V = \left\{ \frac{700}{33} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{3}{4}x+2 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x+\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-24& = & 99x+264 \\\Leftrightarrow & 44x \color{red}{-24} \color{blue}{+24} \color{blue}{-99x} & = & \color{red}{99x} +264 \color{blue}{-99x} \color{blue}{+24} \\\Leftrightarrow & -55x& = & 288 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{288}{-55} \\\Leftrightarrow & x = \frac{-288}{55} & & \\ & V = \left\{ \frac{-288}{55} \right\} & \\\end{align}\)
8. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{9}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 100 }{ \color{blue}{180} })& = & (\frac{-144}{ \color{blue}{180} }x+\frac{1440}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+100& = & -144x+1440 \\\Leftrightarrow & 45x \color{red}{+100} \color{blue}{-100} \color{blue}{+144x} & = & \color{red}{-144x} +1440 \color{blue}{+144x} \color{blue}{-100} \\\Leftrightarrow & 189x& = & 1340 \\\Leftrightarrow & \frac{189x}{ \color{red}{189} }& = & \frac{1340}{189} \\\Leftrightarrow & x = \frac{1340}{189} & & \\ & V = \left\{ \frac{1340}{189} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & -35x-294 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{+35x} & = & \color{red}{-35x} -294 \color{blue}{+35x} \color{blue}{-24} \\\Leftrightarrow & 41x& = & -318 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{-318}{41} \\\Leftrightarrow & x = \frac{-318}{41} & & \\ & V = \left\{ \frac{-318}{41} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x-720 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} -720 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & -680 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-680}{-27} \\\Leftrightarrow & x = \frac{680}{27} & & \\ & V = \left\{ \frac{680}{27} \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 40x-640 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-40x} & = & \color{red}{40x} -640 \color{blue}{-40x} \color{blue}{-15} \\\Leftrightarrow & -24x& = & -655 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-655}{-24} \\\Leftrightarrow & x = \frac{655}{24} & & \\ & V = \left\{ \frac{655}{24} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{4}{5}x-1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x-\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 312x-390 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-312x} & = & \color{red}{312x} -390 \color{blue}{-312x} \color{blue}{+120} \\\Leftrightarrow & -247x& = & -270 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-270}{-247} \\\Leftrightarrow & x = \frac{270}{247} & & \\ & V = \left\{ \frac{270}{247} \right\} & \\\end{align}\)