Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & 6x+60 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{-6x} & = & \color{red}{6x} +60 \color{blue}{-6x} \color{blue}{-25} \\\Leftrightarrow & -x& = & 35 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{35}{-1} \\\Leftrightarrow & x = -35 & & \\ & V = \left\{ -35 \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-8& = & 15x+240 \\\Leftrightarrow & 10x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} +240 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & 248 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{248}{-5} \\\Leftrightarrow & x = \frac{-248}{5} & & \\ & V = \left\{ \frac{-248}{5} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & -112x-400 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{+112x} & = & \color{red}{-112x} -400 \color{blue}{+112x} \color{blue}{+25} \\\Leftrightarrow & 152x& = & -375 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-375}{152} \\\Leftrightarrow & x = \frac{-375}{152} & & \\ & V = \left\{ \frac{-375}{152} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & -80x-96 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{+80x} & = & \color{red}{-80x} -96 \color{blue}{+80x} \color{blue}{-9} \\\Leftrightarrow & 92x& = & -105 \\\Leftrightarrow & \frac{92x}{ \color{red}{92} }& = & \frac{-105}{92} \\\Leftrightarrow & x = \frac{-105}{92} & & \\ & V = \left\{ \frac{-105}{92} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & 36x+240 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{-36x} & = & \color{red}{36x} +240 \color{blue}{-36x} \color{blue}{-4} \\\Leftrightarrow & -21x& = & 236 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{236}{-21} \\\Leftrightarrow & x = \frac{-236}{21} & & \\ & V = \left\{ \frac{-236}{21} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & 396x-1650 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-396x} & = & \color{red}{396x} -1650 \color{blue}{-396x} \color{blue}{-60} \\\Leftrightarrow & -341x& = & -1710 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-1710}{-341} \\\Leftrightarrow & x = \frac{1710}{341} & & \\ & V = \left\{ \frac{1710}{341} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x-\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & -32x-144 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{+32x} & = & \color{red}{-32x} -144 \color{blue}{+32x} \color{blue}{+9} \\\Leftrightarrow & 56x& = & -135 \\\Leftrightarrow & \frac{56x}{ \color{red}{56} }& = & \frac{-135}{56} \\\Leftrightarrow & x = \frac{-135}{56} & & \\ & V = \left\{ \frac{-135}{56} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 14x-168 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-14x} & = & \color{red}{14x} -168 \color{blue}{-14x} \color{blue}{-12} \\\Leftrightarrow & 7x& = & -180 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-180}{7} \\\Leftrightarrow & x = \frac{-180}{7} & & \\ & V = \left\{ \frac{-180}{7} \right\} & \\\end{align}\)
9. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x-\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-50& = & 22x-880 \\\Leftrightarrow & 55x \color{red}{-50} \color{blue}{+50} \color{blue}{-22x} & = & \color{red}{22x} -880 \color{blue}{-22x} \color{blue}{+50} \\\Leftrightarrow & 33x& = & -830 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-830}{33} \\\Leftrightarrow & x = \frac{-830}{33} & & \\ & V = \left\{ \frac{-830}{33} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x+630 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} +630 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & 670 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{670}{-27} \\\Leftrightarrow & x = \frac{-670}{27} & & \\ & V = \left\{ \frac{-670}{27} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{1}{6}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+9& = & 5x+120 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{-5x} & = & \color{red}{5x} +120 \color{blue}{-5x} \color{blue}{-9} \\\Leftrightarrow & x& = & 111 \\ & V = \left\{ 111 \right\} & \\\end{align}\)
12. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x+\frac{378}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & 21x+378 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{-21x} & = & \color{red}{21x} +378 \color{blue}{-21x} \color{blue}{+28} \\\Leftrightarrow & -3x& = & 406 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{406}{-3} \\\Leftrightarrow & x = \frac{-406}{3} & & \\ & V = \left\{ \frac{-406}{3} \right\} & \\\end{align}\)