Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & -80x+240 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{+80x} & = & \color{red}{-80x} +240 \color{blue}{+80x} \color{blue}{+15} \\\Leftrightarrow & 104x& = & 255 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{255}{104} \\\Leftrightarrow & x = \frac{255}{104} & & \\ & V = \left\{ \frac{255}{104} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & -60x+108 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{+60x} & = & \color{red}{-60x} +108 \color{blue}{+60x} \color{blue}{-16} \\\Leftrightarrow & 69x& = & 92 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{92}{69} \\\Leftrightarrow & x = \frac{4}{3} & & \\ & V = \left\{ \frac{4}{3} \right\} & \\\end{align}\)
3. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{5}{4}x+3 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }- \frac{ 112 }{ \color{blue}{420} })& = & (\frac{525}{ \color{blue}{420} }x+\frac{1260}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x-112& = & 525x+1260 \\\Leftrightarrow & 60x \color{red}{-112} \color{blue}{+112} \color{blue}{-525x} & = & \color{red}{525x} +1260 \color{blue}{-525x} \color{blue}{+112} \\\Leftrightarrow & -465x& = & 1372 \\\Leftrightarrow & \frac{-465x}{ \color{red}{-465} }& = & \frac{1372}{-465} \\\Leftrightarrow & x = \frac{-1372}{465} & & \\ & V = \left\{ \frac{-1372}{465} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -294x+1470 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+294x} & = & \color{red}{-294x} +1470 \color{blue}{+294x} \color{blue}{-120} \\\Leftrightarrow & 329x& = & 1350 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{1350}{329} \\\Leftrightarrow & x = \frac{1350}{329} & & \\ & V = \left\{ \frac{1350}{329} \right\} & \\\end{align}\)
5. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{5}{ \color{blue}{10} }x-\frac{10}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x-3& = & 5x-10 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-5x} & = & \color{red}{5x} -10 \color{blue}{-5x} \color{blue}{+3} \\\Leftrightarrow & -3x& = & -7 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-7}{-3} \\\Leftrightarrow & x = \frac{7}{3} & & \\ & V = \left\{ \frac{7}{3} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{-5}{6}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & -25x-180 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{+25x} & = & \color{red}{-25x} -180 \color{blue}{+25x} \color{blue}{+8} \\\Leftrightarrow & 31x& = & -172 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{-172}{31} \\\Leftrightarrow & x = \frac{-172}{31} & & \\ & V = \left\{ \frac{-172}{31} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 100 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+100& = & -112x+420 \\\Leftrightarrow & 35x \color{red}{+100} \color{blue}{-100} \color{blue}{+112x} & = & \color{red}{-112x} +420 \color{blue}{+112x} \color{blue}{-100} \\\Leftrightarrow & 147x& = & 320 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{320}{147} \\\Leftrightarrow & x = \frac{320}{147} & & \\ & V = \left\{ \frac{320}{147} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{3}{5}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & 234x+1950 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{-234x} & = & \color{red}{234x} +1950 \color{blue}{-234x} \color{blue}{-120} \\\Leftrightarrow & -169x& = & 1830 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{1830}{-169} \\\Leftrightarrow & x = \frac{-1830}{169} & & \\ & V = \left\{ \frac{-1830}{169} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x-252 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & -216 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-216}{7} \\\Leftrightarrow & x = \frac{-216}{7} & & \\ & V = \left\{ \frac{-216}{7} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{14}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{56}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+25& = & 56x-350 \\\Leftrightarrow & 35x \color{red}{+25} \color{blue}{-25} \color{blue}{-56x} & = & \color{red}{56x} -350 \color{blue}{-56x} \color{blue}{-25} \\\Leftrightarrow & -21x& = & -375 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-375}{-21} \\\Leftrightarrow & x = \frac{125}{7} & & \\ & V = \left\{ \frac{125}{7} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & -44x-66 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+44x} & = & \color{red}{-44x} -66 \color{blue}{+44x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & -42 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-42}{77} \\\Leftrightarrow & x = \frac{-6}{11} & & \\ & V = \left\{ \frac{-6}{11} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -294x+1050 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+294x} & = & \color{red}{-294x} +1050 \color{blue}{+294x} \color{blue}{-60} \\\Leftrightarrow & 329x& = & 990 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{990}{329} \\\Leftrightarrow & x = \frac{990}{329} & & \\ & V = \left\{ \frac{990}{329} \right\} & \\\end{align}\)