Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-9& = & 7x-42 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-7x} & = & \color{red}{7x} -42 \color{blue}{-7x} \color{blue}{+9} \\\Leftrightarrow & -x& = & -33 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-33}{-1} \\\Leftrightarrow & x = 33 & & \\ & V = \left\{ 33 \right\} & \\\end{align}\)
2. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-147}{ \color{blue}{105} }x-\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-30& = & -147x-105 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{+147x} & = & \color{red}{-147x} -105 \color{blue}{+147x} \color{blue}{+30} \\\Leftrightarrow & 182x& = & -75 \\\Leftrightarrow & \frac{182x}{ \color{red}{182} }& = & \frac{-75}{182} \\\Leftrightarrow & x = \frac{-75}{182} & & \\ & V = \left\{ \frac{-75}{182} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & -56x+84 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{+56x} & = & \color{red}{-56x} +84 \color{blue}{+56x} \color{blue}{-60} \\\Leftrightarrow & 77x& = & 24 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{24}{77} \\\Leftrightarrow & x = \frac{24}{77} & & \\ & V = \left\{ \frac{24}{77} \right\} & \\\end{align}\)
4. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{7}{6}x+8 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 126 }{ \color{blue}{546} })& = & (\frac{637}{ \color{blue}{546} }x+\frac{4368}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+126& = & 637x+4368 \\\Leftrightarrow & 78x \color{red}{+126} \color{blue}{-126} \color{blue}{-637x} & = & \color{red}{637x} +4368 \color{blue}{-637x} \color{blue}{-126} \\\Leftrightarrow & -559x& = & 4242 \\\Leftrightarrow & \frac{-559x}{ \color{red}{-559} }& = & \frac{4242}{-559} \\\Leftrightarrow & x = \frac{-4242}{559} & & \\ & V = \left\{ \frac{-4242}{559} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{14}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+25& = & 84x-490 \\\Leftrightarrow & 35x \color{red}{+25} \color{blue}{-25} \color{blue}{-84x} & = & \color{red}{84x} -490 \color{blue}{-84x} \color{blue}{-25} \\\Leftrightarrow & -49x& = & -515 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-515}{-49} \\\Leftrightarrow & x = \frac{515}{49} & & \\ & V = \left\{ \frac{515}{49} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 154x+462 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-154x} & = & \color{red}{154x} +462 \color{blue}{-154x} \color{blue}{-12} \\\Leftrightarrow & -121x& = & 450 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{450}{-121} \\\Leftrightarrow & x = \frac{-450}{121} & & \\ & V = \left\{ \frac{-450}{121} \right\} & \\\end{align}\)
7. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{4}{3}x+6 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }- \frac{ 35 }{ \color{blue}{63} })& = & (\frac{84}{ \color{blue}{63} }x+\frac{378}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x-35& = & 84x+378 \\\Leftrightarrow & 9x \color{red}{-35} \color{blue}{+35} \color{blue}{-84x} & = & \color{red}{84x} +378 \color{blue}{-84x} \color{blue}{+35} \\\Leftrightarrow & -75x& = & 413 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{413}{-75} \\\Leftrightarrow & x = \frac{-413}{75} & & \\ & V = \left\{ \frac{-413}{75} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 78x+390 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-78x} & = & \color{red}{78x} +390 \color{blue}{-78x} \color{blue}{-60} \\\Leftrightarrow & -13x& = & 330 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{330}{-13} \\\Leftrightarrow & x = \frac{-330}{13} & & \\ & V = \left\{ \frac{-330}{13} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{3}{2}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{135}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 135x-720 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-135x} & = & \color{red}{135x} -720 \color{blue}{-135x} \color{blue}{+20} \\\Leftrightarrow & -117x& = & -700 \\\Leftrightarrow & \frac{-117x}{ \color{red}{-117} }& = & \frac{-700}{-117} \\\Leftrightarrow & x = \frac{700}{117} & & \\ & V = \left\{ \frac{700}{117} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 3, 14 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{70x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 70x-45& = & 42x+840 \\\Leftrightarrow & 70x \color{red}{-45} \color{blue}{+45} \color{blue}{-42x} & = & \color{red}{42x} +840 \color{blue}{-42x} \color{blue}{+45} \\\Leftrightarrow & 28x& = & 885 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{885}{28} \\\Leftrightarrow & x = \frac{885}{28} & & \\ & V = \left\{ \frac{885}{28} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-160}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & -160x+420 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{+160x} & = & \color{red}{-160x} +420 \color{blue}{+160x} \color{blue}{-25} \\\Leftrightarrow & 172x& = & 395 \\\Leftrightarrow & \frac{172x}{ \color{red}{172} }& = & \frac{395}{172} \\\Leftrightarrow & x = \frac{395}{172} & & \\ & V = \left\{ \frac{395}{172} \right\} & \\\end{align}\)
12. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -336x-1440 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+336x} & = & \color{red}{-336x} -1440 \color{blue}{+336x} \color{blue}{-75} \\\Leftrightarrow & 376x& = & -1515 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{-1515}{376} \\\Leftrightarrow & x = \frac{-1515}{376} & & \\ & V = \left\{ \frac{-1515}{376} \right\} & \\\end{align}\)