Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{280 is het kleinste gemene veelvoud van 7, 8 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{8}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{280.} (\frac{40x}{ \color{blue}{280} }+ \frac{ 175 }{ \color{blue}{280} })& = & (\frac{-112}{ \color{blue}{280} }x+\frac{1400}{ \color{blue}{280} }) \color{blue}{.280} \\\Leftrightarrow & 40x+175& = & -112x+1400 \\\Leftrightarrow & 40x \color{red}{+175} \color{blue}{-175} \color{blue}{+112x} & = & \color{red}{-112x} +1400 \color{blue}{+112x} \color{blue}{-175} \\\Leftrightarrow & 152x& = & 1225 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{1225}{152} \\\Leftrightarrow & x = \frac{1225}{152} & & \\ & V = \left\{ \frac{1225}{152} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{1}{3}x-8 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{80}{ \color{blue}{240} }x-\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 80x-1920 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-80x} & = & \color{red}{80x} -1920 \color{blue}{-80x} \color{blue}{-75} \\\Leftrightarrow & -32x& = & -1995 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{-1995}{-32} \\\Leftrightarrow & x = \frac{1995}{32} & & \\ & V = \left\{ \frac{1995}{32} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{5}{4}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & 75x+420 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{-75x} & = & \color{red}{75x} +420 \color{blue}{-75x} \color{blue}{-25} \\\Leftrightarrow & -63x& = & 395 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{395}{-63} \\\Leftrightarrow & x = \frac{-395}{63} & & \\ & V = \left\{ \frac{-395}{63} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x+\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 176x+528 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-176x} & = & \color{red}{176x} +528 \color{blue}{-176x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & 504 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{504}{-143} \\\Leftrightarrow & x = \frac{-504}{143} & & \\ & V = \left\{ \frac{-504}{143} \right\} & \\\end{align}\)
5. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x-84 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} -84 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & -78 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-78}{-5} \\\Leftrightarrow & x = \frac{78}{5} & & \\ & V = \left\{ \frac{78}{5} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{4}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 312x+2340 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-312x} & = & \color{red}{312x} +2340 \color{blue}{-312x} \color{blue}{-90} \\\Leftrightarrow & -247x& = & 2250 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{2250}{-247} \\\Leftrightarrow & x = \frac{-2250}{247} & & \\ & V = \left\{ \frac{-2250}{247} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{5}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 105x+42 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-105x} & = & \color{red}{105x} +42 \color{blue}{-105x} \color{blue}{+24} \\\Leftrightarrow & -91x& = & 66 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{66}{-91} \\\Leftrightarrow & x = \frac{-66}{91} & & \\ & V = \left\{ \frac{-66}{91} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x-\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-48& = & 39x-624 \\\Leftrightarrow & 52x \color{red}{-48} \color{blue}{+48} \color{blue}{-39x} & = & \color{red}{39x} -624 \color{blue}{-39x} \color{blue}{+48} \\\Leftrightarrow & 13x& = & -576 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-576}{13} \\\Leftrightarrow & x = \frac{-576}{13} & & \\ & V = \left\{ \frac{-576}{13} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{7}{2}x-8 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{455}{ \color{blue}{130} }x-\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-30& = & 455x-1040 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-455x} & = & \color{red}{455x} -1040 \color{blue}{-455x} \color{blue}{+30} \\\Leftrightarrow & -429x& = & -1010 \\\Leftrightarrow & \frac{-429x}{ \color{red}{-429} }& = & \frac{-1010}{-429} \\\Leftrightarrow & x = \frac{1010}{429} & & \\ & V = \left\{ \frac{1010}{429} \right\} & \\\end{align}\)
10. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{3}{5}x+3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{72}{ \color{blue}{120} }x+\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & 72x+360 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{-72x} & = & \color{red}{72x} +360 \color{blue}{-72x} \color{blue}{-45} \\\Leftrightarrow & -52x& = & 315 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{315}{-52} \\\Leftrightarrow & x = \frac{-315}{52} & & \\ & V = \left\{ \frac{-315}{52} \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & 96x-240 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{-96x} & = & \color{red}{96x} -240 \color{blue}{-96x} \color{blue}{+25} \\\Leftrightarrow & -76x& = & -215 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{-215}{-76} \\\Leftrightarrow & x = \frac{215}{76} & & \\ & V = \left\{ \frac{215}{76} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{3}{4}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 99x+660 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-99x} & = & \color{red}{99x} +660 \color{blue}{-99x} \color{blue}{+60} \\\Leftrightarrow & -55x& = & 720 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{720}{-55} \\\Leftrightarrow & x = \frac{-144}{11} & & \\ & V = \left\{ \frac{-144}{11} \right\} & \\\end{align}\)