Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 66x+330 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-66x} & = & \color{red}{66x} +330 \color{blue}{-66x} \color{blue}{-120} \\\Leftrightarrow & -11x& = & 210 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{210}{-11} \\\Leftrightarrow & x = \frac{-210}{11} & & \\ & V = \left\{ \frac{-210}{11} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{6}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+120& = & 65x+780 \\\Leftrightarrow & 78x \color{red}{+120} \color{blue}{-120} \color{blue}{-65x} & = & \color{red}{65x} +780 \color{blue}{-65x} \color{blue}{-120} \\\Leftrightarrow & 13x& = & 660 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{660}{13} \\\Leftrightarrow & x = \frac{660}{13} & & \\ & V = \left\{ \frac{660}{13} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{7}{2}x+6 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{42}{ \color{blue}{12} }x+\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 42x+72 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-42x} & = & \color{red}{42x} +72 \color{blue}{-42x} \color{blue}{-5} \\\Leftrightarrow & -38x& = & 67 \\\Leftrightarrow & \frac{-38x}{ \color{red}{-38} }& = & \frac{67}{-38} \\\Leftrightarrow & x = \frac{-67}{38} & & \\ & V = \left\{ \frac{-67}{38} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & -294x-630 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{+294x} & = & \color{red}{-294x} -630 \color{blue}{+294x} \color{blue}{+120} \\\Leftrightarrow & 329x& = & -510 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-510}{329} \\\Leftrightarrow & x = \frac{-510}{329} & & \\ & V = \left\{ \frac{-510}{329} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{48}{ \color{blue}{36} }x-\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & 48x-144 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{-48x} & = & \color{red}{48x} -144 \color{blue}{-48x} \color{blue}{-16} \\\Leftrightarrow & -39x& = & -160 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-160}{-39} \\\Leftrightarrow & x = \frac{160}{39} & & \\ & V = \left\{ \frac{160}{39} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & -117x-312 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{+117x} & = & \color{red}{-117x} -312 \color{blue}{+117x} \color{blue}{+36} \\\Leftrightarrow & 169x& = & -276 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-276}{169} \\\Leftrightarrow & x = \frac{-276}{169} & & \\ & V = \left\{ \frac{-276}{169} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & -48x+120 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{+48x} & = & \color{red}{-48x} +120 \color{blue}{+48x} \color{blue}{+25} \\\Leftrightarrow & 58x& = & 145 \\\Leftrightarrow & \frac{58x}{ \color{red}{58} }& = & \frac{145}{58} \\\Leftrightarrow & x = \frac{5}{2} & & \\ & V = \left\{ \frac{5}{2} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{7}{6}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{98}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & 98x-504 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{-98x} & = & \color{red}{98x} -504 \color{blue}{-98x} \color{blue}{+35} \\\Leftrightarrow & -86x& = & -469 \\\Leftrightarrow & \frac{-86x}{ \color{red}{-86} }& = & \frac{-469}{-86} \\\Leftrightarrow & x = \frac{469}{86} & & \\ & V = \left\{ \frac{469}{86} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -294x+1050 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+294x} & = & \color{red}{-294x} +1050 \color{blue}{+294x} \color{blue}{+90} \\\Leftrightarrow & 329x& = & 1140 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{1140}{329} \\\Leftrightarrow & x = \frac{1140}{329} & & \\ & V = \left\{ \frac{1140}{329} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & -70x+84 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{+70x} & = & \color{red}{-70x} +84 \color{blue}{+70x} \color{blue}{+35} \\\Leftrightarrow & 76x& = & 119 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{119}{76} \\\Leftrightarrow & x = \frac{119}{76} & & \\ & V = \left\{ \frac{119}{76} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & -70x-336 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{+70x} & = & \color{red}{-70x} -336 \color{blue}{+70x} \color{blue}{+9} \\\Leftrightarrow & 91x& = & -327 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-327}{91} \\\Leftrightarrow & x = \frac{-327}{91} & & \\ & V = \left\{ \frac{-327}{91} \right\} & \\\end{align}\)
12. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 105 }{ \color{blue}{455} })& = & (\frac{-364}{ \color{blue}{455} }x-\frac{2730}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+105& = & -364x-2730 \\\Leftrightarrow & 65x \color{red}{+105} \color{blue}{-105} \color{blue}{+364x} & = & \color{red}{-364x} -2730 \color{blue}{+364x} \color{blue}{-105} \\\Leftrightarrow & 429x& = & -2835 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-2835}{429} \\\Leftrightarrow & x = \frac{-945}{143} & & \\ & V = \left\{ \frac{-945}{143} \right\} & \\\end{align}\)