Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{-75}{ \color{blue}{45} }x+\frac{315}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-20& = & -75x+315 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{+75x} & = & \color{red}{-75x} +315 \color{blue}{+75x} \color{blue}{+20} \\\Leftrightarrow & 84x& = & 335 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{335}{84} \\\Leftrightarrow & x = \frac{335}{84} & & \\ & V = \left\{ \frac{335}{84} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-10& = & 9x-126 \\\Leftrightarrow & 6x \color{red}{-10} \color{blue}{+10} \color{blue}{-9x} & = & \color{red}{9x} -126 \color{blue}{-9x} \color{blue}{+10} \\\Leftrightarrow & -3x& = & -116 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-116}{-3} \\\Leftrightarrow & x = \frac{116}{3} & & \\ & V = \left\{ \frac{116}{3} \right\} & \\\end{align}\)
3. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{-5}{6}x-6 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 105 }{ \color{blue}{336} })& = & (\frac{-280}{ \color{blue}{336} }x-\frac{2016}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+105& = & -280x-2016 \\\Leftrightarrow & 48x \color{red}{+105} \color{blue}{-105} \color{blue}{+280x} & = & \color{red}{-280x} -2016 \color{blue}{+280x} \color{blue}{-105} \\\Leftrightarrow & 328x& = & -2121 \\\Leftrightarrow & \frac{328x}{ \color{red}{328} }& = & \frac{-2121}{328} \\\Leftrightarrow & x = \frac{-2121}{328} & & \\ & V = \left\{ \frac{-2121}{328} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{3}{4}x+2 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{117}{ \color{blue}{156} }x+\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+24& = & 117x+312 \\\Leftrightarrow & 52x \color{red}{+24} \color{blue}{-24} \color{blue}{-117x} & = & \color{red}{117x} +312 \color{blue}{-117x} \color{blue}{-24} \\\Leftrightarrow & -65x& = & 288 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{288}{-65} \\\Leftrightarrow & x = \frac{-288}{65} & & \\ & V = \left\{ \frac{-288}{65} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{-7}{4}x-6 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-140}{ \color{blue}{80} }x-\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & -140x-480 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{+140x} & = & \color{red}{-140x} -480 \color{blue}{+140x} \color{blue}{+15} \\\Leftrightarrow & 156x& = & -465 \\\Leftrightarrow & \frac{156x}{ \color{red}{156} }& = & \frac{-465}{156} \\\Leftrightarrow & x = \frac{-155}{52} & & \\ & V = \left\{ \frac{-155}{52} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-400}{ \color{blue}{240} }x+\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & -400x+1680 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{+400x} & = & \color{red}{-400x} +1680 \color{blue}{+400x} \color{blue}{-75} \\\Leftrightarrow & 448x& = & 1605 \\\Leftrightarrow & \frac{448x}{ \color{red}{448} }& = & \frac{1605}{448} \\\Leftrightarrow & x = \frac{1605}{448} & & \\ & V = \left\{ \frac{1605}{448} \right\} & \\\end{align}\)
7. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{4}{ \color{blue}{12} }x+\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+5& = & 4x+96 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-4x} & = & \color{red}{4x} +96 \color{blue}{-4x} \color{blue}{-5} \\\Leftrightarrow & -x& = & 91 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{91}{-1} \\\Leftrightarrow & x = -91 & & \\ & V = \left\{ -91 \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & 44x+660 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-44x} & = & \color{red}{44x} +660 \color{blue}{-44x} \color{blue}{-60} \\\Leftrightarrow & -11x& = & 600 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{600}{-11} \\\Leftrightarrow & x = \frac{-600}{11} & & \\ & V = \left\{ \frac{-600}{11} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & -12x+30 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{+12x} & = & \color{red}{-12x} +30 \color{blue}{+12x} \color{blue}{+4} \\\Leftrightarrow & 27x& = & 34 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{34}{27} \\\Leftrightarrow & x = \frac{34}{27} & & \\ & V = \left\{ \frac{34}{27} \right\} & \\\end{align}\)
10. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{3}{4}x+1 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 16 }{ \color{blue}{28} })& = & (\frac{21}{ \color{blue}{28} }x+\frac{28}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+16& = & 21x+28 \\\Leftrightarrow & 4x \color{red}{+16} \color{blue}{-16} \color{blue}{-21x} & = & \color{red}{21x} +28 \color{blue}{-21x} \color{blue}{-16} \\\Leftrightarrow & -17x& = & 12 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{12}{-17} \\\Leftrightarrow & x = \frac{-12}{17} & & \\ & V = \left\{ \frac{-12}{17} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x+\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-48& = & 39x+468 \\\Leftrightarrow & 52x \color{red}{-48} \color{blue}{+48} \color{blue}{-39x} & = & \color{red}{39x} +468 \color{blue}{-39x} \color{blue}{+48} \\\Leftrightarrow & 13x& = & 516 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{516}{13} \\\Leftrightarrow & x = \frac{516}{13} & & \\ & V = \left\{ \frac{516}{13} \right\} & \\\end{align}\)
12. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }- \frac{ 105 }{ \color{blue}{455} })& = & (\frac{-637}{ \color{blue}{455} }x+\frac{2730}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x-105& = & -637x+2730 \\\Leftrightarrow & 65x \color{red}{-105} \color{blue}{+105} \color{blue}{+637x} & = & \color{red}{-637x} +2730 \color{blue}{+637x} \color{blue}{+105} \\\Leftrightarrow & 702x& = & 2835 \\\Leftrightarrow & \frac{702x}{ \color{red}{702} }& = & \frac{2835}{702} \\\Leftrightarrow & x = \frac{105}{26} & & \\ & V = \left\{ \frac{105}{26} \right\} & \\\end{align}\)