Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{99}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 99x+528 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-99x} & = & \color{red}{99x} +528 \color{blue}{-99x} \color{blue}{-30} \\\Leftrightarrow & -77x& = & 498 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{498}{-77} \\\Leftrightarrow & x = \frac{-498}{77} & & \\ & V = \left\{ \frac{-498}{77} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{1}{4}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & 39x+156 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{-39x} & = & \color{red}{39x} +156 \color{blue}{-39x} \color{blue}{+36} \\\Leftrightarrow & 13x& = & 192 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{192}{13} \\\Leftrightarrow & x = \frac{192}{13} & & \\ & V = \left\{ \frac{192}{13} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-30& = & 39x-468 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-39x} & = & \color{red}{39x} -468 \color{blue}{-39x} \color{blue}{+30} \\\Leftrightarrow & -13x& = & -438 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-438}{-13} \\\Leftrightarrow & x = \frac{438}{13} & & \\ & V = \left\{ \frac{438}{13} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 40 }{ \color{blue}{260} })& = & (\frac{-208}{ \color{blue}{260} }x-\frac{1820}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-40& = & -208x-1820 \\\Leftrightarrow & 65x \color{red}{-40} \color{blue}{+40} \color{blue}{+208x} & = & \color{red}{-208x} -1820 \color{blue}{+208x} \color{blue}{+40} \\\Leftrightarrow & 273x& = & -1780 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-1780}{273} \\\Leftrightarrow & x = \frac{-1780}{273} & & \\ & V = \left\{ \frac{-1780}{273} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-8}{3}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+15& = & -128x+384 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+128x} & = & \color{red}{-128x} +384 \color{blue}{+128x} \color{blue}{-15} \\\Leftrightarrow & 140x& = & 369 \\\Leftrightarrow & \frac{140x}{ \color{red}{140} }& = & \frac{369}{140} \\\Leftrightarrow & x = \frac{369}{140} & & \\ & V = \left\{ \frac{369}{140} \right\} & \\\end{align}\)
6. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 63 }{ \color{blue}{336} })& = & (\frac{-224}{ \color{blue}{336} }x-\frac{2688}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & -224x-2688 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{+224x} & = & \color{red}{-224x} -2688 \color{blue}{+224x} \color{blue}{+63} \\\Leftrightarrow & 272x& = & -2625 \\\Leftrightarrow & \frac{272x}{ \color{red}{272} }& = & \frac{-2625}{272} \\\Leftrightarrow & x = \frac{-2625}{272} & & \\ & V = \left\{ \frac{-2625}{272} \right\} & \\\end{align}\)
7. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{4}{5}x+4 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{96}{ \color{blue}{120} }x+\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 96x+480 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-96x} & = & \color{red}{96x} +480 \color{blue}{-96x} \color{blue}{+75} \\\Leftrightarrow & -76x& = & 555 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{555}{-76} \\\Leftrightarrow & x = \frac{-555}{76} & & \\ & V = \left\{ \frac{-555}{76} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{9}& = & \frac{-5}{6}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-50& = & -75x-90 \\\Leftrightarrow & 18x \color{red}{-50} \color{blue}{+50} \color{blue}{+75x} & = & \color{red}{-75x} -90 \color{blue}{+75x} \color{blue}{+50} \\\Leftrightarrow & 93x& = & -40 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{-40}{93} \\\Leftrightarrow & x = \frac{-40}{93} & & \\ & V = \left\{ \frac{-40}{93} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -84x-1680 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+84x} & = & \color{red}{-84x} -1680 \color{blue}{+84x} \color{blue}{-120} \\\Leftrightarrow & 119x& = & -1800 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1800}{119} \\\Leftrightarrow & x = \frac{-1800}{119} & & \\ & V = \left\{ \frac{-1800}{119} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & 396x+330 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-396x} & = & \color{red}{396x} +330 \color{blue}{-396x} \color{blue}{+60} \\\Leftrightarrow & -341x& = & 390 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{390}{-341} \\\Leftrightarrow & x = \frac{-390}{341} & & \\ & V = \left\{ \frac{-390}{341} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x-\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 48x-720 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-48x} & = & \color{red}{48x} -720 \color{blue}{-48x} \color{blue}{-75} \\\Leftrightarrow & -8x& = & -795 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-795}{-8} \\\Leftrightarrow & x = \frac{795}{8} & & \\ & V = \left\{ \frac{795}{8} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x-\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & -75x-450 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{+75x} & = & \color{red}{-75x} -450 \color{blue}{+75x} \color{blue}{-40} \\\Leftrightarrow & 93x& = & -490 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{-490}{93} \\\Leftrightarrow & x = \frac{-490}{93} & & \\ & V = \left\{ \frac{-490}{93} \right\} & \\\end{align}\)