Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 66x-2640 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-66x} & = & \color{red}{66x} -2640 \color{blue}{-66x} \color{blue}{-120} \\\Leftrightarrow & -11x& = & -2760 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-2760}{-11} \\\Leftrightarrow & x = \frac{2760}{11} & & \\ & V = \left\{ \frac{2760}{11} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{7}{4}x+2 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 36 }{ \color{blue}{132} })& = & (\frac{231}{ \color{blue}{132} }x+\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-36& = & 231x+264 \\\Leftrightarrow & 44x \color{red}{-36} \color{blue}{+36} \color{blue}{-231x} & = & \color{red}{231x} +264 \color{blue}{-231x} \color{blue}{+36} \\\Leftrightarrow & -187x& = & 300 \\\Leftrightarrow & \frac{-187x}{ \color{red}{-187} }& = & \frac{300}{-187} \\\Leftrightarrow & x = \frac{-300}{187} & & \\ & V = \left\{ \frac{-300}{187} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -196x-700 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+196x} & = & \color{red}{-196x} -700 \color{blue}{+196x} \color{blue}{-80} \\\Leftrightarrow & 231x& = & -780 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-780}{231} \\\Leftrightarrow & x = \frac{-260}{77} & & \\ & V = \left\{ \frac{-260}{77} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+18& = & -63x+252 \\\Leftrightarrow & 28x \color{red}{+18} \color{blue}{-18} \color{blue}{+63x} & = & \color{red}{-63x} +252 \color{blue}{+63x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & 234 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{234}{91} \\\Leftrightarrow & x = \frac{18}{7} & & \\ & V = \left\{ \frac{18}{7} \right\} & \\\end{align}\)
5. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{-132}{ \color{blue}{165} }x+\frac{1320}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+75& = & -132x+1320 \\\Leftrightarrow & 55x \color{red}{+75} \color{blue}{-75} \color{blue}{+132x} & = & \color{red}{-132x} +1320 \color{blue}{+132x} \color{blue}{-75} \\\Leftrightarrow & 187x& = & 1245 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{1245}{187} \\\Leftrightarrow & x = \frac{1245}{187} & & \\ & V = \left\{ \frac{1245}{187} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x+420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} +420 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & 480 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{480}{-7} \\\Leftrightarrow & x = \frac{-480}{7} & & \\ & V = \left\{ \frac{-480}{7} \right\} & \\\end{align}\)
7. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{60}{ \color{blue}{36} }x+\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & 60x+36 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-60x} & = & \color{red}{60x} +36 \color{blue}{-60x} \color{blue}{+8} \\\Leftrightarrow & -51x& = & 44 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{44}{-51} \\\Leftrightarrow & x = \frac{-44}{51} & & \\ & V = \left\{ \frac{-44}{51} \right\} & \\\end{align}\)
8. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 100 }{ \color{blue}{260} })& = & (\frac{156}{ \color{blue}{260} }x+\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+100& = & 156x+520 \\\Leftrightarrow & 65x \color{red}{+100} \color{blue}{-100} \color{blue}{-156x} & = & \color{red}{156x} +520 \color{blue}{-156x} \color{blue}{-100} \\\Leftrightarrow & -91x& = & 420 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{420}{-91} \\\Leftrightarrow & x = \frac{-60}{13} & & \\ & V = \left\{ \frac{-60}{13} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{7}{2}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{455}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 455x+650 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-455x} & = & \color{red}{455x} +650 \color{blue}{-455x} \color{blue}{-40} \\\Leftrightarrow & -429x& = & 610 \\\Leftrightarrow & \frac{-429x}{ \color{red}{-429} }& = & \frac{610}{-429} \\\Leftrightarrow & x = \frac{-610}{429} & & \\ & V = \left\{ \frac{-610}{429} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & -273x+624 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{+273x} & = & \color{red}{-273x} +624 \color{blue}{+273x} \color{blue}{+36} \\\Leftrightarrow & 325x& = & 660 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{660}{325} \\\Leftrightarrow & x = \frac{132}{65} & & \\ & V = \left\{ \frac{132}{65} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -126x-270 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+126x} & = & \color{red}{-126x} -270 \color{blue}{+126x} \color{blue}{-20} \\\Leftrightarrow & 141x& = & -290 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-290}{141} \\\Leftrightarrow & x = \frac{-290}{141} & & \\ & V = \left\{ \frac{-290}{141} \right\} & \\\end{align}\)
12. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{-105}{ \color{blue}{126} }x+\frac{756}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & -105x+756 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{+105x} & = & \color{red}{-105x} +756 \color{blue}{+105x} \color{blue}{+70} \\\Leftrightarrow & 123x& = & 826 \\\Leftrightarrow & \frac{123x}{ \color{red}{123} }& = & \frac{826}{123} \\\Leftrightarrow & x = \frac{826}{123} & & \\ & V = \left\{ \frac{826}{123} \right\} & \\\end{align}\)