Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x-\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-15& = & 32x-96 \\\Leftrightarrow & 6x \color{red}{-15} \color{blue}{+15} \color{blue}{-32x} & = & \color{red}{32x} -96 \color{blue}{-32x} \color{blue}{+15} \\\Leftrightarrow & -26x& = & -81 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-81}{-26} \\\Leftrightarrow & x = \frac{81}{26} & & \\ & V = \left\{ \frac{81}{26} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 7x-112 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-7x} & = & \color{red}{7x} -112 \color{blue}{-7x} \color{blue}{-6} \\\Leftrightarrow & -5x& = & -118 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-118}{-5} \\\Leftrightarrow & x = \frac{118}{5} & & \\ & V = \left\{ \frac{118}{5} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & -30x-126 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+30x} & = & \color{red}{-30x} -126 \color{blue}{+30x} \color{blue}{-8} \\\Leftrightarrow & 39x& = & -134 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-134}{39} \\\Leftrightarrow & x = \frac{-134}{39} & & \\ & V = \left\{ \frac{-134}{39} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & -168x-1470 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+168x} & = & \color{red}{-168x} -1470 \color{blue}{+168x} \color{blue}{+45} \\\Leftrightarrow & 203x& = & -1425 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1425}{203} \\\Leftrightarrow & x = \frac{-1425}{203} & & \\ & V = \left\{ \frac{-1425}{203} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{4}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 312x-2730 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-312x} & = & \color{red}{312x} -2730 \color{blue}{-312x} \color{blue}{-60} \\\Leftrightarrow & -247x& = & -2790 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-2790}{-247} \\\Leftrightarrow & x = \frac{2790}{247} & & \\ & V = \left\{ \frac{2790}{247} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 20x+120 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-20x} & = & \color{red}{20x} +120 \color{blue}{-20x} \color{blue}{-16} \\\Leftrightarrow & -5x& = & 104 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{104}{-5} \\\Leftrightarrow & x = \frac{-104}{5} & & \\ & V = \left\{ \frac{-104}{5} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{7}{4}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{245}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-40& = & 245x-140 \\\Leftrightarrow & 28x \color{red}{-40} \color{blue}{+40} \color{blue}{-245x} & = & \color{red}{245x} -140 \color{blue}{-245x} \color{blue}{+40} \\\Leftrightarrow & -217x& = & -100 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-100}{-217} \\\Leftrightarrow & x = \frac{100}{217} & & \\ & V = \left\{ \frac{100}{217} \right\} & \\\end{align}\)
8. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x+\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x-75& = & -168x+600 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{+168x} & = & \color{red}{-168x} +600 \color{blue}{+168x} \color{blue}{+75} \\\Leftrightarrow & 208x& = & 675 \\\Leftrightarrow & \frac{208x}{ \color{red}{208} }& = & \frac{675}{208} \\\Leftrightarrow & x = \frac{675}{208} & & \\ & V = \left\{ \frac{675}{208} \right\} & \\\end{align}\)
9. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 56 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{1092}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-56& = & 91x-1092 \\\Leftrightarrow & 26x \color{red}{-56} \color{blue}{+56} \color{blue}{-91x} & = & \color{red}{91x} -1092 \color{blue}{-91x} \color{blue}{+56} \\\Leftrightarrow & -65x& = & -1036 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-1036}{-65} \\\Leftrightarrow & x = \frac{1036}{65} & & \\ & V = \left\{ \frac{1036}{65} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{2}{3}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{104}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & 104x+624 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-104x} & = & \color{red}{104x} +624 \color{blue}{-104x} \color{blue}{+24} \\\Leftrightarrow & -65x& = & 648 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{648}{-65} \\\Leftrightarrow & x = \frac{-648}{65} & & \\ & V = \left\{ \frac{-648}{65} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{7}{2}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{168}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 168x-192 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-168x} & = & \color{red}{168x} -192 \color{blue}{-168x} \color{blue}{+9} \\\Leftrightarrow & -152x& = & -183 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{-183}{-152} \\\Leftrightarrow & x = \frac{183}{152} & & \\ & V = \left\{ \frac{183}{152} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -168x+840 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+168x} & = & \color{red}{-168x} +840 \color{blue}{+168x} \color{blue}{-90} \\\Leftrightarrow & 203x& = & 750 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{750}{203} \\\Leftrightarrow & x = \frac{750}{203} & & \\ & V = \left\{ \frac{750}{203} \right\} & \\\end{align}\)