Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 112 }{ \color{blue}{420} })& = & (\frac{315}{ \color{blue}{420} }x-\frac{2940}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+112& = & 315x-2940 \\\Leftrightarrow & 60x \color{red}{+112} \color{blue}{-112} \color{blue}{-315x} & = & \color{red}{315x} -2940 \color{blue}{-315x} \color{blue}{-112} \\\Leftrightarrow & -255x& = & -3052 \\\Leftrightarrow & \frac{-255x}{ \color{red}{-255} }& = & \frac{-3052}{-255} \\\Leftrightarrow & x = \frac{3052}{255} & & \\ & V = \left\{ \frac{3052}{255} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x-\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & -110x-462 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+110x} & = & \color{red}{-110x} -462 \color{blue}{+110x} \color{blue}{-24} \\\Leftrightarrow & 143x& = & -486 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-486}{143} \\\Leftrightarrow & x = \frac{-486}{143} & & \\ & V = \left\{ \frac{-486}{143} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 60 }{ \color{blue}{165} })& = & (\frac{33}{ \color{blue}{165} }x+\frac{1155}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-60& = & 33x+1155 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-33x} & = & \color{red}{33x} +1155 \color{blue}{-33x} \color{blue}{+60} \\\Leftrightarrow & 22x& = & 1215 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{1215}{22} \\\Leftrightarrow & x = \frac{1215}{22} & & \\ & V = \left\{ \frac{1215}{22} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -72x+90 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+72x} & = & \color{red}{-72x} +90 \color{blue}{+72x} \color{blue}{-40} \\\Leftrightarrow & 87x& = & 50 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{50}{87} \\\Leftrightarrow & x = \frac{50}{87} & & \\ & V = \left\{ \frac{50}{87} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 12x-300 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-12x} & = & \color{red}{12x} -300 \color{blue}{-12x} \color{blue}{-16} \\\Leftrightarrow & 3x& = & -316 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-316}{3} \\\Leftrightarrow & x = \frac{-316}{3} & & \\ & V = \left\{ \frac{-316}{3} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{7}{2}x+5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x+\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 49x+70 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-49x} & = & \color{red}{49x} +70 \color{blue}{-49x} \color{blue}{-6} \\\Leftrightarrow & -47x& = & 64 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{64}{-47} \\\Leftrightarrow & x = \frac{-64}{47} & & \\ & V = \left\{ \frac{-64}{47} \right\} & \\\end{align}\)
7. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{105}{ \color{blue}{45} }x-\frac{270}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-10& = & 105x-270 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-105x} & = & \color{red}{105x} -270 \color{blue}{-105x} \color{blue}{+10} \\\Leftrightarrow & -96x& = & -260 \\\Leftrightarrow & \frac{-96x}{ \color{red}{-96} }& = & \frac{-260}{-96} \\\Leftrightarrow & x = \frac{65}{24} & & \\ & V = \left\{ \frac{65}{24} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{273}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-18& = & 273x-546 \\\Leftrightarrow & 26x \color{red}{-18} \color{blue}{+18} \color{blue}{-273x} & = & \color{red}{273x} -546 \color{blue}{-273x} \color{blue}{+18} \\\Leftrightarrow & -247x& = & -528 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-528}{-247} \\\Leftrightarrow & x = \frac{528}{247} & & \\ & V = \left\{ \frac{528}{247} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & 20x-420 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{-20x} & = & \color{red}{20x} -420 \color{blue}{-20x} \color{blue}{-18} \\\Leftrightarrow & -5x& = & -438 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-438}{-5} \\\Leftrightarrow & x = \frac{438}{5} & & \\ & V = \left\{ \frac{438}{5} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 175x+490 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-175x} & = & \color{red}{175x} +490 \color{blue}{-175x} \color{blue}{+40} \\\Leftrightarrow & -161x& = & 530 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{530}{-161} \\\Leftrightarrow & x = \frac{-530}{161} & & \\ & V = \left\{ \frac{-530}{161} \right\} & \\\end{align}\)
11. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-104}{ \color{blue}{260} }x+\frac{1040}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & -104x+1040 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{+104x} & = & \color{red}{-104x} +1040 \color{blue}{+104x} \color{blue}{+80} \\\Leftrightarrow & 169x& = & 1120 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{1120}{169} \\\Leftrightarrow & x = \frac{1120}{169} & & \\ & V = \left\{ \frac{1120}{169} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{7}{2}x-1 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{280}{ \color{blue}{80} }x-\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 280x-80 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-280x} & = & \color{red}{280x} -80 \color{blue}{-280x} \color{blue}{-15} \\\Leftrightarrow & -264x& = & -95 \\\Leftrightarrow & \frac{-264x}{ \color{red}{-264} }& = & \frac{-95}{-264} \\\Leftrightarrow & x = \frac{95}{264} & & \\ & V = \left\{ \frac{95}{264} \right\} & \\\end{align}\)