Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}+\frac{5}{16}=\frac{3}{4}x+5\)
2. \(\frac{x}{3}-\frac{3}{7}=\frac{5}{2}x-4\)
3. \(\frac{x}{3}+\frac{4}{13}=\frac{-7}{4}x-7\)
4. \(\frac{x}{3}-\frac{4}{7}=\frac{-3}{4}x-5\)
5. \(\frac{x}{2}-\frac{4}{7}=\frac{-7}{5}x+5\)
6. \(\frac{x}{2}-\frac{2}{9}=\frac{1}{3}x-5\)
7. \(\frac{x}{7}-\frac{2}{7}=\frac{-3}{4}x-1\)
8. \(\frac{x}{7}-\frac{3}{14}=\frac{3}{4}x+8\)
9. \(\frac{x}{3}+\frac{2}{15}=\frac{5}{4}x+8\)
10. \(\frac{x}{2}+\frac{2}{15}=\frac{7}{3}x-5\)
11. \(\frac{x}{2}+\frac{2}{7}=\frac{-5}{3}x+4\)
12. \(\frac{x}{3}+\frac{5}{6}=\frac{-4}{5}x-6\)

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{3}{4}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{60}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 60x+400 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-60x} & = & \color{red}{60x} +400 \color{blue}{-60x} \color{blue}{-25} \\\Leftrightarrow & -44x& = & 375 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{375}{-44} \\\Leftrightarrow & x = \frac{-375}{44} & & \\ & V = \left\{ \frac{-375}{44} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 105x-168 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-105x} & = & \color{red}{105x} -168 \color{blue}{-105x} \color{blue}{+18} \\\Leftrightarrow & -91x& = & -150 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-150}{-91} \\\Leftrightarrow & x = \frac{150}{91} & & \\ & V = \left\{ \frac{150}{91} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x-\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+48& = & -273x-1092 \\\Leftrightarrow & 52x \color{red}{+48} \color{blue}{-48} \color{blue}{+273x} & = & \color{red}{-273x} -1092 \color{blue}{+273x} \color{blue}{-48} \\\Leftrightarrow & 325x& = & -1140 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-1140}{325} \\\Leftrightarrow & x = \frac{-228}{65} & & \\ & V = \left\{ \frac{-228}{65} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-48& = & -63x-420 \\\Leftrightarrow & 28x \color{red}{-48} \color{blue}{+48} \color{blue}{+63x} & = & \color{red}{-63x} -420 \color{blue}{+63x} \color{blue}{+48} \\\Leftrightarrow & 91x& = & -372 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-372}{91} \\\Leftrightarrow & x = \frac{-372}{91} & & \\ & V = \left\{ \frac{-372}{91} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & -98x+350 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+98x} & = & \color{red}{-98x} +350 \color{blue}{+98x} \color{blue}{+40} \\\Leftrightarrow & 133x& = & 390 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{390}{133} \\\Leftrightarrow & x = \frac{390}{133} & & \\ & V = \left\{ \frac{390}{133} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x-\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & 6x-90 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} -90 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & 3x& = & -86 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-86}{3} \\\Leftrightarrow & x = \frac{-86}{3} & & \\ & V = \left\{ \frac{-86}{3} \right\} & \\\end{align}\)
7. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 8 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x-\frac{28}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-8& = & -21x-28 \\\Leftrightarrow & 4x \color{red}{-8} \color{blue}{+8} \color{blue}{+21x} & = & \color{red}{-21x} -28 \color{blue}{+21x} \color{blue}{+8} \\\Leftrightarrow & 25x& = & -20 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{-20}{25} \\\Leftrightarrow & x = \frac{-4}{5} & & \\ & V = \left\{ \frac{-4}{5} \right\} & \\\end{align}\)
8. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{3}{4}x+8 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 6 }{ \color{blue}{28} })& = & (\frac{21}{ \color{blue}{28} }x+\frac{224}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-6& = & 21x+224 \\\Leftrightarrow & 4x \color{red}{-6} \color{blue}{+6} \color{blue}{-21x} & = & \color{red}{21x} +224 \color{blue}{-21x} \color{blue}{+6} \\\Leftrightarrow & -17x& = & 230 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{230}{-17} \\\Leftrightarrow & x = \frac{-230}{17} & & \\ & V = \left\{ \frac{-230}{17} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{5}{4}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+8& = & 75x+480 \\\Leftrightarrow & 20x \color{red}{+8} \color{blue}{-8} \color{blue}{-75x} & = & \color{red}{75x} +480 \color{blue}{-75x} \color{blue}{-8} \\\Leftrightarrow & -55x& = & 472 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{472}{-55} \\\Leftrightarrow & x = \frac{-472}{55} & & \\ & V = \left\{ \frac{-472}{55} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & 70x-150 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{-70x} & = & \color{red}{70x} -150 \color{blue}{-70x} \color{blue}{-4} \\\Leftrightarrow & -55x& = & -154 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-154}{-55} \\\Leftrightarrow & x = \frac{14}{5} & & \\ & V = \left\{ \frac{14}{5} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -70x+168 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+70x} & = & \color{red}{-70x} +168 \color{blue}{+70x} \color{blue}{-12} \\\Leftrightarrow & 91x& = & 156 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{156}{91} \\\Leftrightarrow & x = \frac{12}{7} & & \\ & V = \left\{ \frac{12}{7} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 3, 6 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+25& = & -24x-180 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} -180 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 34x& = & -205 \\\Leftrightarrow & \frac{34x}{ \color{red}{34} }& = & \frac{-205}{34} \\\Leftrightarrow & x = \frac{-205}{34} & & \\ & V = \left\{ \frac{-205}{34} \right\} & \\\end{align}\)