Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 5x+150 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-5x} & = & \color{red}{5x} +150 \color{blue}{-5x} \color{blue}{-8} \\\Leftrightarrow & x& = & 142 \\ & V = \left\{ 142 \right\} & \\\end{align}\)
2. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 30 }{ \color{blue}{195} })& = & (\frac{455}{ \color{blue}{195} }x-\frac{1560}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-30& = & 455x-1560 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{-455x} & = & \color{red}{455x} -1560 \color{blue}{-455x} \color{blue}{+30} \\\Leftrightarrow & -416x& = & -1530 \\\Leftrightarrow & \frac{-416x}{ \color{red}{-416} }& = & \frac{-1530}{-416} \\\Leftrightarrow & x = \frac{765}{208} & & \\ & V = \left\{ \frac{765}{208} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x-490 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} -490 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & -530 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-530}{-21} \\\Leftrightarrow & x = \frac{530}{21} & & \\ & V = \left\{ \frac{530}{21} \right\} & \\\end{align}\)
4. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x+\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & 16x+480 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} +480 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & 4x& = & 465 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{465}{4} \\\Leftrightarrow & x = \frac{465}{4} & & \\ & V = \left\{ \frac{465}{4} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & -140x-504 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{+140x} & = & \color{red}{-140x} -504 \color{blue}{+140x} \color{blue}{+36} \\\Leftrightarrow & 161x& = & -468 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-468}{161} \\\Leftrightarrow & x = \frac{-468}{161} & & \\ & V = \left\{ \frac{-468}{161} \right\} & \\\end{align}\)
6. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 40 }{ \color{blue}{220} })& = & (\frac{264}{ \color{blue}{220} }x-\frac{220}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+40& = & 264x-220 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{-264x} & = & \color{red}{264x} -220 \color{blue}{-264x} \color{blue}{-40} \\\Leftrightarrow & -209x& = & -260 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-260}{-209} \\\Leftrightarrow & x = \frac{260}{209} & & \\ & V = \left\{ \frac{260}{209} \right\} & \\\end{align}\)
7. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 25 }{ \color{blue}{35} })& = & (\frac{42}{ \color{blue}{35} }x-\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+25& = & 42x-70 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{-42x} & = & \color{red}{42x} -70 \color{blue}{-42x} \color{blue}{-25} \\\Leftrightarrow & -37x& = & -95 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{-95}{-37} \\\Leftrightarrow & x = \frac{95}{37} & & \\ & V = \left\{ \frac{95}{37} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 7, 6 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 175 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+175& = & -84x-630 \\\Leftrightarrow & 30x \color{red}{+175} \color{blue}{-175} \color{blue}{+84x} & = & \color{red}{-84x} -630 \color{blue}{+84x} \color{blue}{-175} \\\Leftrightarrow & 114x& = & -805 \\\Leftrightarrow & \frac{114x}{ \color{red}{114} }& = & \frac{-805}{114} \\\Leftrightarrow & x = \frac{-805}{114} & & \\ & V = \left\{ \frac{-805}{114} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x-420 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} -420 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & -440 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-440}{-21} \\\Leftrightarrow & x = \frac{440}{21} & & \\ & V = \left\{ \frac{440}{21} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x+\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -36x+90 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+36x} & = & \color{red}{-36x} +90 \color{blue}{+36x} \color{blue}{-40} \\\Leftrightarrow & 51x& = & 50 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{50}{51} \\\Leftrightarrow & x = \frac{50}{51} & & \\ & V = \left\{ \frac{50}{51} \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x-\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & -64x-480 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{+64x} & = & \color{red}{-64x} -480 \color{blue}{+64x} \color{blue}{-25} \\\Leftrightarrow & 104x& = & -505 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{-505}{104} \\\Leftrightarrow & x = \frac{-505}{104} & & \\ & V = \left\{ \frac{-505}{104} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -56x-336 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+56x} & = & \color{red}{-56x} -336 \color{blue}{+56x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & -312 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-312}{77} \\\Leftrightarrow & x = \frac{-312}{77} & & \\ & V = \left\{ \frac{-312}{77} \right\} & \\\end{align}\)