Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{189}{ \color{blue}{126} }x+\frac{126}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & 189x+126 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{-189x} & = & \color{red}{189x} +126 \color{blue}{-189x} \color{blue}{+28} \\\Leftrightarrow & -171x& = & 154 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{154}{-171} \\\Leftrightarrow & x = \frac{-154}{171} & & \\ & V = \left\{ \frac{-154}{171} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & 36x-210 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{-36x} & = & \color{red}{36x} -210 \color{blue}{-36x} \color{blue}{+9} \\\Leftrightarrow & -31x& = & -201 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-201}{-31} \\\Leftrightarrow & x = \frac{201}{31} & & \\ & V = \left\{ \frac{201}{31} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+25& = & -98x+140 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{+98x} & = & \color{red}{-98x} +140 \color{blue}{+98x} \color{blue}{-25} \\\Leftrightarrow & 108x& = & 115 \\\Leftrightarrow & \frac{108x}{ \color{red}{108} }& = & \frac{115}{108} \\\Leftrightarrow & x = \frac{115}{108} & & \\ & V = \left\{ \frac{115}{108} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -546x+2340 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+546x} & = & \color{red}{-546x} +2340 \color{blue}{+546x} \color{blue}{-120} \\\Leftrightarrow & 611x& = & 2220 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{2220}{611} \\\Leftrightarrow & x = \frac{2220}{611} & & \\ & V = \left\{ \frac{2220}{611} \right\} & \\\end{align}\)
5. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x+\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-75& = & 55x+330 \\\Leftrightarrow & 33x \color{red}{-75} \color{blue}{+75} \color{blue}{-55x} & = & \color{red}{55x} +330 \color{blue}{-55x} \color{blue}{+75} \\\Leftrightarrow & -22x& = & 405 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{405}{-22} \\\Leftrightarrow & x = \frac{-405}{22} & & \\ & V = \left\{ \frac{-405}{22} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{3}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & 63x-420 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{-63x} & = & \color{red}{63x} -420 \color{blue}{-63x} \color{blue}{-48} \\\Leftrightarrow & -35x& = & -468 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-468}{-35} \\\Leftrightarrow & x = \frac{468}{35} & & \\ & V = \left\{ \frac{468}{35} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+16& = & 15x+240 \\\Leftrightarrow & 20x \color{red}{+16} \color{blue}{-16} \color{blue}{-15x} & = & \color{red}{15x} +240 \color{blue}{-15x} \color{blue}{-16} \\\Leftrightarrow & 5x& = & 224 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{224}{5} \\\Leftrightarrow & x = \frac{224}{5} & & \\ & V = \left\{ \frac{224}{5} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{7}{6}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{455}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 455x-3120 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-455x} & = & \color{red}{455x} -3120 \color{blue}{-455x} \color{blue}{-60} \\\Leftrightarrow & -377x& = & -3180 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{-3180}{-377} \\\Leftrightarrow & x = \frac{3180}{377} & & \\ & V = \left\{ \frac{3180}{377} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & -24x+108 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+24x} & = & \color{red}{-24x} +108 \color{blue}{+24x} \color{blue}{-8} \\\Leftrightarrow & 33x& = & 100 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{100}{33} \\\Leftrightarrow & x = \frac{100}{33} & & \\ & V = \left\{ \frac{100}{33} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & 252x+1680 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{-252x} & = & \color{red}{252x} +1680 \color{blue}{-252x} \color{blue}{+150} \\\Leftrightarrow & -217x& = & 1830 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{1830}{-217} \\\Leftrightarrow & x = \frac{-1830}{217} & & \\ & V = \left\{ \frac{-1830}{217} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{45}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 45x+30 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-45x} & = & \color{red}{45x} +30 \color{blue}{-45x} \color{blue}{+8} \\\Leftrightarrow & -39x& = & 38 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{38}{-39} \\\Leftrightarrow & x = \frac{-38}{39} & & \\ & V = \left\{ \frac{-38}{39} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-40& = & -104x+650 \\\Leftrightarrow & 65x \color{red}{-40} \color{blue}{+40} \color{blue}{+104x} & = & \color{red}{-104x} +650 \color{blue}{+104x} \color{blue}{+40} \\\Leftrightarrow & 169x& = & 690 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{690}{169} \\\Leftrightarrow & x = \frac{690}{169} & & \\ & V = \left\{ \frac{690}{169} \right\} & \\\end{align}\)