Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{4}{3}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & 176x-792 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-176x} & = & \color{red}{176x} -792 \color{blue}{-176x} \color{blue}{-60} \\\Leftrightarrow & -143x& = & -852 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-852}{-143} \\\Leftrightarrow & x = \frac{852}{143} & & \\ & V = \left\{ \frac{852}{143} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{4}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+25& = & 24x-180 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{-24x} & = & \color{red}{24x} -180 \color{blue}{-24x} \color{blue}{-25} \\\Leftrightarrow & -9x& = & -205 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-205}{-9} \\\Leftrightarrow & x = \frac{205}{9} & & \\ & V = \left\{ \frac{205}{9} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x+\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+36& = & 39x+312 \\\Leftrightarrow & 52x \color{red}{+36} \color{blue}{-36} \color{blue}{-39x} & = & \color{red}{39x} +312 \color{blue}{-39x} \color{blue}{-36} \\\Leftrightarrow & 13x& = & 276 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{276}{13} \\\Leftrightarrow & x = \frac{276}{13} & & \\ & V = \left\{ \frac{276}{13} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -28x-168 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+28x} & = & \color{red}{-28x} -168 \color{blue}{+28x} \color{blue}{+24} \\\Leftrightarrow & 49x& = & -144 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-144}{49} \\\Leftrightarrow & x = \frac{-144}{49} & & \\ & V = \left\{ \frac{-144}{49} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 7, 6 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{-7}{4}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 70 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-70& = & -147x+504 \\\Leftrightarrow & 12x \color{red}{-70} \color{blue}{+70} \color{blue}{+147x} & = & \color{red}{-147x} +504 \color{blue}{+147x} \color{blue}{+70} \\\Leftrightarrow & 159x& = & 574 \\\Leftrightarrow & \frac{159x}{ \color{red}{159} }& = & \frac{574}{159} \\\Leftrightarrow & x = \frac{574}{159} & & \\ & V = \left\{ \frac{574}{159} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 50 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+50& = & -56x-140 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{+56x} & = & \color{red}{-56x} -140 \color{blue}{+56x} \color{blue}{-50} \\\Leftrightarrow & 91x& = & -190 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-190}{91} \\\Leftrightarrow & x = \frac{-190}{91} & & \\ & V = \left\{ \frac{-190}{91} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{5}{3}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 80x-96 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-80x} & = & \color{red}{80x} -96 \color{blue}{-80x} \color{blue}{-15} \\\Leftrightarrow & -56x& = & -111 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{-111}{-56} \\\Leftrightarrow & x = \frac{111}{56} & & \\ & V = \left\{ \frac{111}{56} \right\} & \\\end{align}\)
8. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & 6x+18 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{-6x} & = & \color{red}{6x} +18 \color{blue}{-6x} \color{blue}{-10} \\\Leftrightarrow & 3x& = & 8 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{8}{3} \\\Leftrightarrow & x = \frac{8}{3} & & \\ & V = \left\{ \frac{8}{3} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-18& = & 33x+66 \\\Leftrightarrow & 22x \color{red}{-18} \color{blue}{+18} \color{blue}{-33x} & = & \color{red}{33x} +66 \color{blue}{-33x} \color{blue}{+18} \\\Leftrightarrow & -11x& = & 84 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{84}{-11} \\\Leftrightarrow & x = \frac{-84}{11} & & \\ & V = \left\{ \frac{-84}{11} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 56 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-56& = & 105x-1680 \\\Leftrightarrow & 30x \color{red}{-56} \color{blue}{+56} \color{blue}{-105x} & = & \color{red}{105x} -1680 \color{blue}{-105x} \color{blue}{+56} \\\Leftrightarrow & -75x& = & -1624 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-1624}{-75} \\\Leftrightarrow & x = \frac{1624}{75} & & \\ & V = \left\{ \frac{1624}{75} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x+\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & -192x+1440 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{+192x} & = & \color{red}{-192x} +1440 \color{blue}{+192x} \color{blue}{+45} \\\Leftrightarrow & 232x& = & 1485 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{1485}{232} \\\Leftrightarrow & x = \frac{1485}{232} & & \\ & V = \left\{ \frac{1485}{232} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -72x-540 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+72x} & = & \color{red}{-72x} -540 \color{blue}{+72x} \color{blue}{+40} \\\Leftrightarrow & 87x& = & -500 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{-500}{87} \\\Leftrightarrow & x = \frac{-500}{87} & & \\ & V = \left\{ \frac{-500}{87} \right\} & \\\end{align}\)