Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+9& = & 105x-168 \\\Leftrightarrow & 14x \color{red}{+9} \color{blue}{-9} \color{blue}{-105x} & = & \color{red}{105x} -168 \color{blue}{-105x} \color{blue}{-9} \\\Leftrightarrow & -91x& = & -177 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-177}{-91} \\\Leftrightarrow & x = \frac{177}{91} & & \\ & V = \left\{ \frac{177}{91} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{42}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x-42 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} -42 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -34 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-34}{-5} \\\Leftrightarrow & x = \frac{34}{5} & & \\ & V = \left\{ \frac{34}{5} \right\} & \\\end{align}\)
3. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }- \frac{ 2 }{ \color{blue}{15} })& = & (\frac{-25}{ \color{blue}{15} }x+\frac{45}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x-2& = & -25x+45 \\\Leftrightarrow & 3x \color{red}{-2} \color{blue}{+2} \color{blue}{+25x} & = & \color{red}{-25x} +45 \color{blue}{+25x} \color{blue}{+2} \\\Leftrightarrow & 28x& = & 47 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{47}{28} \\\Leftrightarrow & x = \frac{47}{28} & & \\ & V = \left\{ \frac{47}{28} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-80& = & 35x-560 \\\Leftrightarrow & 28x \color{red}{-80} \color{blue}{+80} \color{blue}{-35x} & = & \color{red}{35x} -560 \color{blue}{-35x} \color{blue}{+80} \\\Leftrightarrow & -7x& = & -480 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-480}{-7} \\\Leftrightarrow & x = \frac{480}{7} & & \\ & V = \left\{ \frac{480}{7} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-168 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -144 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-144}{-7} \\\Leftrightarrow & x = \frac{144}{7} & & \\ & V = \left\{ \frac{144}{7} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 147x-126 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-147x} & = & \color{red}{147x} -126 \color{blue}{-147x} \color{blue}{-18} \\\Leftrightarrow & -133x& = & -144 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-144}{-133} \\\Leftrightarrow & x = \frac{144}{133} & & \\ & V = \left\{ \frac{144}{133} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & -80x+240 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{+80x} & = & \color{red}{-80x} +240 \color{blue}{+80x} \color{blue}{-15} \\\Leftrightarrow & 104x& = & 225 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{225}{104} \\\Leftrightarrow & x = \frac{225}{104} & & \\ & V = \left\{ \frac{225}{104} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & 108x+720 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-108x} & = & \color{red}{108x} +720 \color{blue}{-108x} \color{blue}{+20} \\\Leftrightarrow & -93x& = & 740 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{740}{-93} \\\Leftrightarrow & x = \frac{-740}{93} & & \\ & V = \left\{ \frac{-740}{93} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -52x-78 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+52x} & = & \color{red}{-52x} -78 \color{blue}{+52x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & -96 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-96}{91} \\\Leftrightarrow & x = \frac{-96}{91} & & \\ & V = \left\{ \frac{-96}{91} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & -325x-1950 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{+325x} & = & \color{red}{-325x} -1950 \color{blue}{+325x} \color{blue}{-60} \\\Leftrightarrow & 403x& = & -2010 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{-2010}{403} \\\Leftrightarrow & x = \frac{-2010}{403} & & \\ & V = \left\{ \frac{-2010}{403} \right\} & \\\end{align}\)
11. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 175 }{ \color{blue}{385} })& = & (\frac{-539}{ \color{blue}{385} }x+\frac{1540}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+175& = & -539x+1540 \\\Leftrightarrow & 55x \color{red}{+175} \color{blue}{-175} \color{blue}{+539x} & = & \color{red}{-539x} +1540 \color{blue}{+539x} \color{blue}{-175} \\\Leftrightarrow & 594x& = & 1365 \\\Leftrightarrow & \frac{594x}{ \color{red}{594} }& = & \frac{1365}{594} \\\Leftrightarrow & x = \frac{455}{198} & & \\ & V = \left\{ \frac{455}{198} \right\} & \\\end{align}\)
12. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{112}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 56x-112 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-56x} & = & \color{red}{56x} -112 \color{blue}{-56x} \color{blue}{-35} \\\Leftrightarrow & -40x& = & -147 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-147}{-40} \\\Leftrightarrow & x = \frac{147}{40} & & \\ & V = \left\{ \frac{147}{40} \right\} & \\\end{align}\)