Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}+\frac{3}{7}=\frac{3}{2}x+4\)
2. \(\frac{x}{5}+\frac{3}{7}=\frac{1}{2}x+8\)
3. \(\frac{x}{6}-\frac{5}{6}=\frac{-7}{5}x-2\)
4. \(\frac{x}{7}-\frac{5}{11}=\frac{7}{4}x+2\)
5. \(\frac{x}{2}+\frac{5}{16}=\frac{6}{5}x+1\)
6. \(\frac{x}{4}+\frac{5}{6}=\frac{-4}{5}x-5\)
7. \(\frac{x}{7}-\frac{5}{6}=\frac{7}{3}x+2\)
8. \(\frac{x}{5}-\frac{2}{9}=\frac{-2}{3}x+7\)
9. \(\frac{x}{5}+\frac{5}{14}=\frac{-5}{6}x-3\)
10. \(\frac{x}{3}-\frac{3}{16}=\frac{1}{2}x-6\)
11. \(\frac{x}{6}+\frac{5}{11}=\frac{-7}{5}x+8\)
12. \(\frac{x}{3}+\frac{4}{7}=\frac{1}{4}x+1\)

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{3}{2}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 63x+168 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-63x} & = & \color{red}{63x} +168 \color{blue}{-63x} \color{blue}{-18} \\\Leftrightarrow & -49x& = & 150 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{150}{-49} \\\Leftrightarrow & x = \frac{-150}{49} & & \\ & V = \left\{ \frac{-150}{49} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 35x+560 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-35x} & = & \color{red}{35x} +560 \color{blue}{-35x} \color{blue}{-30} \\\Leftrightarrow & -21x& = & 530 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{530}{-21} \\\Leftrightarrow & x = \frac{-530}{21} & & \\ & V = \left\{ \frac{-530}{21} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -42x-60 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+42x} & = & \color{red}{-42x} -60 \color{blue}{+42x} \color{blue}{+25} \\\Leftrightarrow & 47x& = & -35 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-35}{47} \\\Leftrightarrow & x = \frac{-35}{47} & & \\ & V = \left\{ \frac{-35}{47} \right\} & \\\end{align}\)
4. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{7}{4}x+2 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }- \frac{ 140 }{ \color{blue}{308} })& = & (\frac{539}{ \color{blue}{308} }x+\frac{616}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x-140& = & 539x+616 \\\Leftrightarrow & 44x \color{red}{-140} \color{blue}{+140} \color{blue}{-539x} & = & \color{red}{539x} +616 \color{blue}{-539x} \color{blue}{+140} \\\Leftrightarrow & -495x& = & 756 \\\Leftrightarrow & \frac{-495x}{ \color{red}{-495} }& = & \frac{756}{-495} \\\Leftrightarrow & x = \frac{-84}{55} & & \\ & V = \left\{ \frac{-84}{55} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x+\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & 96x+80 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{-96x} & = & \color{red}{96x} +80 \color{blue}{-96x} \color{blue}{-25} \\\Leftrightarrow & -56x& = & 55 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{55}{-56} \\\Leftrightarrow & x = \frac{-55}{56} & & \\ & V = \left\{ \frac{-55}{56} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+50& = & -48x-300 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{+48x} & = & \color{red}{-48x} -300 \color{blue}{+48x} \color{blue}{-50} \\\Leftrightarrow & 63x& = & -350 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{-350}{63} \\\Leftrightarrow & x = \frac{-50}{9} & & \\ & V = \left\{ \frac{-50}{9} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 98x+84 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-98x} & = & \color{red}{98x} +84 \color{blue}{-98x} \color{blue}{+35} \\\Leftrightarrow & -92x& = & 119 \\\Leftrightarrow & \frac{-92x}{ \color{red}{-92} }& = & \frac{119}{-92} \\\Leftrightarrow & x = \frac{-119}{92} & & \\ & V = \left\{ \frac{-119}{92} \right\} & \\\end{align}\)
8. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{-30}{ \color{blue}{45} }x+\frac{315}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-10& = & -30x+315 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{+30x} & = & \color{red}{-30x} +315 \color{blue}{+30x} \color{blue}{+10} \\\Leftrightarrow & 39x& = & 325 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{325}{39} \\\Leftrightarrow & x = \frac{25}{3} & & \\ & V = \left\{ \frac{25}{3} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{-5}{6}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+75& = & -175x-630 \\\Leftrightarrow & 42x \color{red}{+75} \color{blue}{-75} \color{blue}{+175x} & = & \color{red}{-175x} -630 \color{blue}{+175x} \color{blue}{-75} \\\Leftrightarrow & 217x& = & -705 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-705}{217} \\\Leftrightarrow & x = \frac{-705}{217} & & \\ & V = \left\{ \frac{-705}{217} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 24x-288 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-24x} & = & \color{red}{24x} -288 \color{blue}{-24x} \color{blue}{+9} \\\Leftrightarrow & -8x& = & -279 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-279}{-8} \\\Leftrightarrow & x = \frac{279}{8} & & \\ & V = \left\{ \frac{279}{8} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & -462x+2640 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{+462x} & = & \color{red}{-462x} +2640 \color{blue}{+462x} \color{blue}{-150} \\\Leftrightarrow & 517x& = & 2490 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{2490}{517} \\\Leftrightarrow & x = \frac{2490}{517} & & \\ & V = \left\{ \frac{2490}{517} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{4}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & 21x+84 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{-48} \\\Leftrightarrow & 7x& = & 36 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{36}{7} \\\Leftrightarrow & x = \frac{36}{7} & & \\ & V = \left\{ \frac{36}{7} \right\} & \\\end{align}\)