Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & -462x-1320 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+462x} & = & \color{red}{-462x} -1320 \color{blue}{+462x} \color{blue}{+60} \\\Leftrightarrow & 517x& = & -1260 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-1260}{517} \\\Leftrightarrow & x = \frac{-1260}{517} & & \\ & V = \left\{ \frac{-1260}{517} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{56}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 56x+140 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-56x} & = & \color{red}{56x} +140 \color{blue}{-56x} \color{blue}{+21} \\\Leftrightarrow & -46x& = & 161 \\\Leftrightarrow & \frac{-46x}{ \color{red}{-46} }& = & \frac{161}{-46} \\\Leftrightarrow & x = \frac{-7}{2} & & \\ & V = \left\{ \frac{-7}{2} \right\} & \\\end{align}\)
3. \(\text{40 is het kleinste gemene veelvoud van 2, 8 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{40.} (\frac{20x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{8}{ \color{blue}{40} }x+\frac{200}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 20x+25& = & 8x+200 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{-8x} & = & \color{red}{8x} +200 \color{blue}{-8x} \color{blue}{-25} \\\Leftrightarrow & 12x& = & 175 \\\Leftrightarrow & \frac{12x}{ \color{red}{12} }& = & \frac{175}{12} \\\Leftrightarrow & x = \frac{175}{12} & & \\ & V = \left\{ \frac{175}{12} \right\} & \\\end{align}\)
4. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-78}{ \color{blue}{195} }x-\frac{780}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & -78x-780 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{+78x} & = & \color{red}{-78x} -780 \color{blue}{+78x} \color{blue}{-45} \\\Leftrightarrow & 143x& = & -825 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-825}{143} \\\Leftrightarrow & x = \frac{-75}{13} & & \\ & V = \left\{ \frac{-75}{13} \right\} & \\\end{align}\)
5. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 20 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x+\frac{28}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-20& = & -21x+28 \\\Leftrightarrow & 4x \color{red}{-20} \color{blue}{+20} \color{blue}{+21x} & = & \color{red}{-21x} +28 \color{blue}{+21x} \color{blue}{+20} \\\Leftrightarrow & 25x& = & 48 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{48}{25} \\\Leftrightarrow & x = \frac{48}{25} & & \\ & V = \left\{ \frac{48}{25} \right\} & \\\end{align}\)
6. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{182}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & 182x+78 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-182x} & = & \color{red}{182x} +78 \color{blue}{-182x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & 54 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{54}{-143} \\\Leftrightarrow & x = \frac{-54}{143} & & \\ & V = \left\{ \frac{-54}{143} \right\} & \\\end{align}\)
7. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 84 }{ \color{blue}{273} })& = & (\frac{91}{ \color{blue}{273} }x+\frac{1638}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-84& = & 91x+1638 \\\Leftrightarrow & 39x \color{red}{-84} \color{blue}{+84} \color{blue}{-91x} & = & \color{red}{91x} +1638 \color{blue}{-91x} \color{blue}{+84} \\\Leftrightarrow & -52x& = & 1722 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{1722}{-52} \\\Leftrightarrow & x = \frac{-861}{26} & & \\ & V = \left\{ \frac{-861}{26} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & -84x-420 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{+84x} & = & \color{red}{-84x} -420 \color{blue}{+84x} \color{blue}{+150} \\\Leftrightarrow & 119x& = & -270 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-270}{119} \\\Leftrightarrow & x = \frac{-270}{119} & & \\ & V = \left\{ \frac{-270}{119} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-8& = & -45x-120 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{+45x} & = & \color{red}{-45x} -120 \color{blue}{+45x} \color{blue}{+8} \\\Leftrightarrow & 57x& = & -112 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-112}{57} \\\Leftrightarrow & x = \frac{-112}{57} & & \\ & V = \left\{ \frac{-112}{57} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & -546x+1560 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{+546x} & = & \color{red}{-546x} +1560 \color{blue}{+546x} \color{blue}{+120} \\\Leftrightarrow & 611x& = & 1680 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{1680}{611} \\\Leftrightarrow & x = \frac{1680}{611} & & \\ & V = \left\{ \frac{1680}{611} \right\} & \\\end{align}\)
11. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{5}{ \color{blue}{10} }x-\frac{30}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x+3& = & 5x-30 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-5x} & = & \color{red}{5x} -30 \color{blue}{-5x} \color{blue}{-3} \\\Leftrightarrow & -3x& = & -33 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-33}{-3} \\\Leftrightarrow & x = 11 & & \\ & V = \left\{ 11 \right\} & \\\end{align}\)
12. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{2}{3}x+6 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{24}{ \color{blue}{36} }x+\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & 24x+216 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-24x} & = & \color{red}{24x} +216 \color{blue}{-24x} \color{blue}{-8} \\\Leftrightarrow & -15x& = & 208 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{208}{-15} \\\Leftrightarrow & x = \frac{-208}{15} & & \\ & V = \left\{ \frac{-208}{15} \right\} & \\\end{align}\)