Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-60& = & -88x-528 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{+88x} & = & \color{red}{-88x} -528 \color{blue}{+88x} \color{blue}{+60} \\\Leftrightarrow & 121x& = & -468 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-468}{121} \\\Leftrightarrow & x = \frac{-468}{121} & & \\ & V = \left\{ \frac{-468}{121} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{4}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-8& = & 15x-480 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} -480 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & -472 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-472}{-3} \\\Leftrightarrow & x = \frac{472}{3} & & \\ & V = \left\{ \frac{472}{3} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+36& = & -63x-420 \\\Leftrightarrow & 28x \color{red}{+36} \color{blue}{-36} \color{blue}{+63x} & = & \color{red}{-63x} -420 \color{blue}{+63x} \color{blue}{-36} \\\Leftrightarrow & 91x& = & -456 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-456}{91} \\\Leftrightarrow & x = \frac{-456}{91} & & \\ & V = \left\{ \frac{-456}{91} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{28}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & 28x-210 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{-28x} & = & \color{red}{28x} -210 \color{blue}{-28x} \color{blue}{+20} \\\Leftrightarrow & 7x& = & -190 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-190}{7} \\\Leftrightarrow & x = \frac{-190}{7} & & \\ & V = \left\{ \frac{-190}{7} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x+210 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} +210 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 186 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{186}{-7} \\\Leftrightarrow & x = \frac{-186}{7} & & \\ & V = \left\{ \frac{-186}{7} \right\} & \\\end{align}\)
6. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 6 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x+\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-6& = & 7x+84 \\\Leftrightarrow & 4x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} +84 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -3x& = & 90 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{90}{-3} \\\Leftrightarrow & x = -30 & & \\ & V = \left\{ -30 \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-8}{3}x-5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x-\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -176x-330 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+176x} & = & \color{red}{-176x} -330 \color{blue}{+176x} \color{blue}{-30} \\\Leftrightarrow & 209x& = & -360 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-360}{209} \\\Leftrightarrow & x = \frac{-360}{209} & & \\ & V = \left\{ \frac{-360}{209} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{5}{4}x-3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{45}{ \color{blue}{36} }x-\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-16& = & 45x-108 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{-45x} & = & \color{red}{45x} -108 \color{blue}{-45x} \color{blue}{+16} \\\Leftrightarrow & -33x& = & -92 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-92}{-33} \\\Leftrightarrow & x = \frac{92}{33} & & \\ & V = \left\{ \frac{92}{33} \right\} & \\\end{align}\)
9. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{-7}{4}x+2 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-21}{ \color{blue}{12} }x+\frac{24}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-10& = & -21x+24 \\\Leftrightarrow & 4x \color{red}{-10} \color{blue}{+10} \color{blue}{+21x} & = & \color{red}{-21x} +24 \color{blue}{+21x} \color{blue}{+10} \\\Leftrightarrow & 25x& = & 34 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{34}{25} \\\Leftrightarrow & x = \frac{34}{25} & & \\ & V = \left\{ \frac{34}{25} \right\} & \\\end{align}\)
10. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }- \frac{ 140 }{ \color{blue}{455} })& = & (\frac{-637}{ \color{blue}{455} }x+\frac{2275}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x-140& = & -637x+2275 \\\Leftrightarrow & 65x \color{red}{-140} \color{blue}{+140} \color{blue}{+637x} & = & \color{red}{-637x} +2275 \color{blue}{+637x} \color{blue}{+140} \\\Leftrightarrow & 702x& = & 2415 \\\Leftrightarrow & \frac{702x}{ \color{red}{702} }& = & \frac{2415}{702} \\\Leftrightarrow & x = \frac{805}{234} & & \\ & V = \left\{ \frac{805}{234} \right\} & \\\end{align}\)
11. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 25 }{ \color{blue}{35} })& = & (\frac{42}{ \color{blue}{35} }x+\frac{245}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+25& = & 42x+245 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{-42x} & = & \color{red}{42x} +245 \color{blue}{-42x} \color{blue}{-25} \\\Leftrightarrow & -37x& = & 220 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{220}{-37} \\\Leftrightarrow & x = \frac{-220}{37} & & \\ & V = \left\{ \frac{-220}{37} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -546x+780 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+546x} & = & \color{red}{-546x} +780 \color{blue}{+546x} \color{blue}{-120} \\\Leftrightarrow & 611x& = & 660 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{660}{611} \\\Leftrightarrow & x = \frac{660}{611} & & \\ & V = \left\{ \frac{660}{611} \right\} & \\\end{align}\)