Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{-5}{6}x+5 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & -200x+1200 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{+200x} & = & \color{red}{-200x} +1200 \color{blue}{+200x} \color{blue}{+75} \\\Leftrightarrow & 248x& = & 1275 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{1275}{248} \\\Leftrightarrow & x = \frac{1275}{248} & & \\ & V = \left\{ \frac{1275}{248} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 18 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+18& = & 196x+672 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{-196x} & = & \color{red}{196x} +672 \color{blue}{-196x} \color{blue}{-18} \\\Leftrightarrow & -175x& = & 654 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{654}{-175} \\\Leftrightarrow & x = \frac{-654}{175} & & \\ & V = \left\{ \frac{-654}{175} \right\} & \\\end{align}\)
3. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{5}{2}x-3 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{120}{ \color{blue}{48} }x-\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 120x-144 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-120x} & = & \color{red}{120x} -144 \color{blue}{-120x} \color{blue}{-9} \\\Leftrightarrow & -104x& = & -153 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{-153}{-104} \\\Leftrightarrow & x = \frac{153}{104} & & \\ & V = \left\{ \frac{153}{104} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{14}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 30 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-30& = & 28x+84 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-28x} & = & \color{red}{28x} +84 \color{blue}{-28x} \color{blue}{+30} \\\Leftrightarrow & -7x& = & 114 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{114}{-7} \\\Leftrightarrow & x = \frac{-114}{7} & & \\ & V = \left\{ \frac{-114}{7} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{4}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 168x-1260 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-168x} & = & \color{red}{168x} -1260 \color{blue}{-168x} \color{blue}{-150} \\\Leftrightarrow & -133x& = & -1410 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-1410}{-133} \\\Leftrightarrow & x = \frac{1410}{133} & & \\ & V = \left\{ \frac{1410}{133} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{7}{4}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-18& = & 105x+60 \\\Leftrightarrow & 20x \color{red}{-18} \color{blue}{+18} \color{blue}{-105x} & = & \color{red}{105x} +60 \color{blue}{-105x} \color{blue}{+18} \\\Leftrightarrow & -85x& = & 78 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{78}{-85} \\\Leftrightarrow & x = \frac{-78}{85} & & \\ & V = \left\{ \frac{-78}{85} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+40& = & -126x+450 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{+126x} & = & \color{red}{-126x} +450 \color{blue}{+126x} \color{blue}{-40} \\\Leftrightarrow & 171x& = & 410 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{410}{171} \\\Leftrightarrow & x = \frac{410}{171} & & \\ & V = \left\{ \frac{410}{171} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x+\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & -112x+640 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{+112x} & = & \color{red}{-112x} +640 \color{blue}{+112x} \color{blue}{-25} \\\Leftrightarrow & 132x& = & 615 \\\Leftrightarrow & \frac{132x}{ \color{red}{132} }& = & \frac{615}{132} \\\Leftrightarrow & x = \frac{205}{44} & & \\ & V = \left\{ \frac{205}{44} \right\} & \\\end{align}\)
9. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 40x-400 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-40x} & = & \color{red}{40x} -400 \color{blue}{-40x} \color{blue}{-15} \\\Leftrightarrow & -24x& = & -415 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-415}{-24} \\\Leftrightarrow & x = \frac{415}{24} & & \\ & V = \left\{ \frac{415}{24} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x+\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & 12x+36 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{-12x} & = & \color{red}{12x} +36 \color{blue}{-12x} \color{blue}{-16} \\\Leftrightarrow & -3x& = & 20 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{20}{-3} \\\Leftrightarrow & x = \frac{-20}{3} & & \\ & V = \left\{ \frac{-20}{3} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -12x-30 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+12x} & = & \color{red}{-12x} -30 \color{blue}{+12x} \color{blue}{-9} \\\Leftrightarrow & 17x& = & -39 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-39}{17} \\\Leftrightarrow & x = \frac{-39}{17} & & \\ & V = \left\{ \frac{-39}{17} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-352}{ \color{blue}{132} }x-\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-24& = & -352x-264 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+352x} & = & \color{red}{-352x} -264 \color{blue}{+352x} \color{blue}{+24} \\\Leftrightarrow & 385x& = & -240 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{-240}{385} \\\Leftrightarrow & x = \frac{-48}{77} & & \\ & V = \left\{ \frac{-48}{77} \right\} & \\\end{align}\)