Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 100 }{ \color{blue}{260} })& = & (\frac{-364}{ \color{blue}{260} }x-\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+100& = & -364x-1300 \\\Leftrightarrow & 65x \color{red}{+100} \color{blue}{-100} \color{blue}{+364x} & = & \color{red}{-364x} -1300 \color{blue}{+364x} \color{blue}{-100} \\\Leftrightarrow & 429x& = & -1400 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-1400}{429} \\\Leftrightarrow & x = \frac{-1400}{429} & & \\ & V = \left\{ \frac{-1400}{429} \right\} & \\\end{align}\)
2. \(\text{20 is het kleinste gemene veelvoud van 5, 10 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{3}{4}x+4 \\\Leftrightarrow & \color{blue}{20.} (\frac{4x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{15}{ \color{blue}{20} }x+\frac{80}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 4x+6& = & 15x+80 \\\Leftrightarrow & 4x \color{red}{+6} \color{blue}{-6} \color{blue}{-15x} & = & \color{red}{15x} +80 \color{blue}{-15x} \color{blue}{-6} \\\Leftrightarrow & -11x& = & 74 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{74}{-11} \\\Leftrightarrow & x = \frac{-74}{11} & & \\ & V = \left\{ \frac{-74}{11} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-24& = & 7x-168 \\\Leftrightarrow & 6x \color{red}{-24} \color{blue}{+24} \color{blue}{-7x} & = & \color{red}{7x} -168 \color{blue}{-7x} \color{blue}{+24} \\\Leftrightarrow & -x& = & -144 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-144}{-1} \\\Leftrightarrow & x = 144 & & \\ & V = \left\{ 144 \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & -84x-210 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{+84x} & = & \color{red}{-84x} -210 \color{blue}{+84x} \color{blue}{+75} \\\Leftrightarrow & 119x& = & -135 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-135}{119} \\\Leftrightarrow & x = \frac{-135}{119} & & \\ & V = \left\{ \frac{-135}{119} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{5}{2}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 175x+560 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-175x} & = & \color{red}{175x} +560 \color{blue}{-175x} \color{blue}{+40} \\\Leftrightarrow & -161x& = & 600 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{600}{-161} \\\Leftrightarrow & x = \frac{-600}{161} & & \\ & V = \left\{ \frac{-600}{161} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{5}{6}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 25x-120 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-25x} & = & \color{red}{25x} -120 \color{blue}{-25x} \color{blue}{-4} \\\Leftrightarrow & -19x& = & -124 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-124}{-19} \\\Leftrightarrow & x = \frac{124}{19} & & \\ & V = \left\{ \frac{124}{19} \right\} & \\\end{align}\)
7. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-8}{ \color{blue}{20} }x+\frac{40}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & -8x+40 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{+8x} & = & \color{red}{-8x} +40 \color{blue}{+8x} \color{blue}{-6} \\\Leftrightarrow & 13x& = & 34 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{34}{13} \\\Leftrightarrow & x = \frac{34}{13} & & \\ & V = \left\{ \frac{34}{13} \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{336}{ \color{blue}{240} }x+\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 336x+240 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-336x} & = & \color{red}{336x} +240 \color{blue}{-336x} \color{blue}{+45} \\\Leftrightarrow & -296x& = & 285 \\\Leftrightarrow & \frac{-296x}{ \color{red}{-296} }& = & \frac{285}{-296} \\\Leftrightarrow & x = \frac{-285}{296} & & \\ & V = \left\{ \frac{-285}{296} \right\} & \\\end{align}\)
9. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{8}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }- \frac{ 35 }{ \color{blue}{56} })& = & (\frac{14}{ \color{blue}{56} }x-\frac{336}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x-35& = & 14x-336 \\\Leftrightarrow & 8x \color{red}{-35} \color{blue}{+35} \color{blue}{-14x} & = & \color{red}{14x} -336 \color{blue}{-14x} \color{blue}{+35} \\\Leftrightarrow & -6x& = & -301 \\\Leftrightarrow & \frac{-6x}{ \color{red}{-6} }& = & \frac{-301}{-6} \\\Leftrightarrow & x = \frac{301}{6} & & \\ & V = \left\{ \frac{301}{6} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{7}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{462}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 462x+2640 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-462x} & = & \color{red}{462x} +2640 \color{blue}{-462x} \color{blue}{-120} \\\Leftrightarrow & -407x& = & 2520 \\\Leftrightarrow & \frac{-407x}{ \color{red}{-407} }& = & \frac{2520}{-407} \\\Leftrightarrow & x = \frac{-2520}{407} & & \\ & V = \left\{ \frac{-2520}{407} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x+108 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} +108 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & 104 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{104}{-3} \\\Leftrightarrow & x = \frac{-104}{3} & & \\ & V = \left\{ \frac{-104}{3} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & 108x+720 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-108x} & = & \color{red}{108x} +720 \color{blue}{-108x} \color{blue}{+50} \\\Leftrightarrow & -93x& = & 770 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{770}{-93} \\\Leftrightarrow & x = \frac{-770}{93} & & \\ & V = \left\{ \frac{-770}{93} \right\} & \\\end{align}\)