Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-3}{4}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+80& = & -105x-560 \\\Leftrightarrow & 28x \color{red}{+80} \color{blue}{-80} \color{blue}{+105x} & = & \color{red}{-105x} -560 \color{blue}{+105x} \color{blue}{-80} \\\Leftrightarrow & 133x& = & -640 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-640}{133} \\\Leftrightarrow & x = \frac{-640}{133} & & \\ & V = \left\{ \frac{-640}{133} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 147x-252 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-147x} & = & \color{red}{147x} -252 \color{blue}{-147x} \color{blue}{-12} \\\Leftrightarrow & -133x& = & -264 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-264}{-133} \\\Leftrightarrow & x = \frac{264}{133} & & \\ & V = \left\{ \frac{264}{133} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-18& = & -63x+252 \\\Leftrightarrow & 28x \color{red}{-18} \color{blue}{+18} \color{blue}{+63x} & = & \color{red}{-63x} +252 \color{blue}{+63x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & 270 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{270}{91} \\\Leftrightarrow & x = \frac{270}{91} & & \\ & V = \left\{ \frac{270}{91} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{156}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & 156x-130 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{-156x} & = & \color{red}{156x} -130 \color{blue}{-156x} \color{blue}{-20} \\\Leftrightarrow & -91x& = & -150 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-150}{-91} \\\Leftrightarrow & x = \frac{150}{91} & & \\ & V = \left\{ \frac{150}{91} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 3, 14 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{70x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 70x-75& = & 294x+1470 \\\Leftrightarrow & 70x \color{red}{-75} \color{blue}{+75} \color{blue}{-294x} & = & \color{red}{294x} +1470 \color{blue}{-294x} \color{blue}{+75} \\\Leftrightarrow & -224x& = & 1545 \\\Leftrightarrow & \frac{-224x}{ \color{red}{-224} }& = & \frac{1545}{-224} \\\Leftrightarrow & x = \frac{-1545}{224} & & \\ & V = \left\{ \frac{-1545}{224} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & -64x-400 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{+64x} & = & \color{red}{-64x} -400 \color{blue}{+64x} \color{blue}{-25} \\\Leftrightarrow & 104x& = & -425 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{-425}{104} \\\Leftrightarrow & x = \frac{-425}{104} & & \\ & V = \left\{ \frac{-425}{104} \right\} & \\\end{align}\)
7. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 30 }{ \color{blue}{165} })& = & (\frac{231}{ \color{blue}{165} }x-\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-30& = & 231x-660 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{-231x} & = & \color{red}{231x} -660 \color{blue}{-231x} \color{blue}{+30} \\\Leftrightarrow & -176x& = & -630 \\\Leftrightarrow & \frac{-176x}{ \color{red}{-176} }& = & \frac{-630}{-176} \\\Leftrightarrow & x = \frac{315}{88} & & \\ & V = \left\{ \frac{315}{88} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -12x+60 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+12x} & = & \color{red}{-12x} +60 \color{blue}{+12x} \color{blue}{-25} \\\Leftrightarrow & 17x& = & 35 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{35}{17} \\\Leftrightarrow & x = \frac{35}{17} & & \\ & V = \left\{ \frac{35}{17} \right\} & \\\end{align}\)
9. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-60}{ \color{blue}{80} }x+\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & -60x+80 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{+60x} & = & \color{red}{-60x} +80 \color{blue}{+60x} \color{blue}{-15} \\\Leftrightarrow & 76x& = & 65 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{65}{76} \\\Leftrightarrow & x = \frac{65}{76} & & \\ & V = \left\{ \frac{65}{76} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 42x-1050 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-42x} & = & \color{red}{42x} -1050 \color{blue}{-42x} \color{blue}{-60} \\\Leftrightarrow & -7x& = & -1110 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-1110}{-7} \\\Leftrightarrow & x = \frac{1110}{7} & & \\ & V = \left\{ \frac{1110}{7} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{7}& = & \frac{-7}{4}x-8 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 100 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x-\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+100& = & -245x-1120 \\\Leftrightarrow & 28x \color{red}{+100} \color{blue}{-100} \color{blue}{+245x} & = & \color{red}{-245x} -1120 \color{blue}{+245x} \color{blue}{-100} \\\Leftrightarrow & 273x& = & -1220 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-1220}{273} \\\Leftrightarrow & x = \frac{-1220}{273} & & \\ & V = \left\{ \frac{-1220}{273} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-9& = & -80x-384 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{+80x} & = & \color{red}{-80x} -384 \color{blue}{+80x} \color{blue}{+9} \\\Leftrightarrow & 92x& = & -375 \\\Leftrightarrow & \frac{92x}{ \color{red}{92} }& = & \frac{-375}{92} \\\Leftrightarrow & x = \frac{-375}{92} & & \\ & V = \left\{ \frac{-375}{92} \right\} & \\\end{align}\)