Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+15& = & -32x+192 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+32x} & = & \color{red}{-32x} +192 \color{blue}{+32x} \color{blue}{-15} \\\Leftrightarrow & 44x& = & 177 \\\Leftrightarrow & \frac{44x}{ \color{red}{44} }& = & \frac{177}{44} \\\Leftrightarrow & x = \frac{177}{44} & & \\ & V = \left\{ \frac{177}{44} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{-7}{4}x-4 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-42}{ \color{blue}{24} }x-\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & -42x-96 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{+42x} & = & \color{red}{-42x} -96 \color{blue}{+42x} \color{blue}{+15} \\\Leftrightarrow & 50x& = & -81 \\\Leftrightarrow & \frac{50x}{ \color{red}{50} }& = & \frac{-81}{50} \\\Leftrightarrow & x = \frac{-81}{50} & & \\ & V = \left\{ \frac{-81}{50} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x-\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-45& = & -84x-105 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+84x} & = & \color{red}{-84x} -105 \color{blue}{+84x} \color{blue}{+45} \\\Leftrightarrow & 119x& = & -60 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-60}{119} \\\Leftrightarrow & x = \frac{-60}{119} & & \\ & V = \left\{ \frac{-60}{119} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{4}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 56x+126 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-56x} & = & \color{red}{56x} +126 \color{blue}{-56x} \color{blue}{-12} \\\Leftrightarrow & -35x& = & 114 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{114}{-35} \\\Leftrightarrow & x = \frac{-114}{35} & & \\ & V = \left\{ \frac{-114}{35} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & -100x+480 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{+100x} & = & \color{red}{-100x} +480 \color{blue}{+100x} \color{blue}{-18} \\\Leftrightarrow & 115x& = & 462 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{462}{115} \\\Leftrightarrow & x = \frac{462}{115} & & \\ & V = \left\{ \frac{462}{115} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{-8}{3}x-5 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x-\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+10& = & -32x-60 \\\Leftrightarrow & 3x \color{red}{+10} \color{blue}{-10} \color{blue}{+32x} & = & \color{red}{-32x} -60 \color{blue}{+32x} \color{blue}{-10} \\\Leftrightarrow & 35x& = & -70 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{-70}{35} \\\Leftrightarrow & x = -2 & & \\ & V = \left\{ -2 \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & 108x+540 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{-108x} & = & \color{red}{108x} +540 \color{blue}{-108x} \color{blue}{-20} \\\Leftrightarrow & -63x& = & 520 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{520}{-63} \\\Leftrightarrow & x = \frac{-520}{63} & & \\ & V = \left\{ \frac{-520}{63} \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 60 }{ \color{blue}{165} })& = & (\frac{33}{ \color{blue}{165} }x+\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+60& = & 33x+660 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-33x} & = & \color{red}{33x} +660 \color{blue}{-33x} \color{blue}{-60} \\\Leftrightarrow & 22x& = & 600 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{600}{22} \\\Leftrightarrow & x = \frac{300}{11} & & \\ & V = \left\{ \frac{300}{11} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & 52x+1092 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-52x} & = & \color{red}{52x} +1092 \color{blue}{-52x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & 1116 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1116}{-13} \\\Leftrightarrow & x = \frac{-1116}{13} & & \\ & V = \left\{ \frac{-1116}{13} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-336 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -336 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -360 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-360}{-7} \\\Leftrightarrow & x = \frac{360}{7} & & \\ & V = \left\{ \frac{360}{7} \right\} & \\\end{align}\)
11. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{3}{2}x+2 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{195}{ \color{blue}{130} }x+\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-30& = & 195x+260 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-195x} & = & \color{red}{195x} +260 \color{blue}{-195x} \color{blue}{+30} \\\Leftrightarrow & -169x& = & 290 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{290}{-169} \\\Leftrightarrow & x = \frac{-290}{169} & & \\ & V = \left\{ \frac{-290}{169} \right\} & \\\end{align}\)
12. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }- \frac{ 105 }{ \color{blue}{455} })& = & (\frac{-637}{ \color{blue}{455} }x-\frac{3185}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x-105& = & -637x-3185 \\\Leftrightarrow & 65x \color{red}{-105} \color{blue}{+105} \color{blue}{+637x} & = & \color{red}{-637x} -3185 \color{blue}{+637x} \color{blue}{+105} \\\Leftrightarrow & 702x& = & -3080 \\\Leftrightarrow & \frac{702x}{ \color{red}{702} }& = & \frac{-3080}{702} \\\Leftrightarrow & x = \frac{-1540}{351} & & \\ & V = \left\{ \frac{-1540}{351} \right\} & \\\end{align}\)