Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}-\frac{2}{11}=\frac{7}{4}x+6\)
2. \(\frac{x}{3}+\frac{5}{16}=\frac{1}{2}x+5\)
3. \(\frac{x}{7}-\frac{3}{11}=\frac{-2}{5}x-3\)
4. \(\frac{x}{7}-\frac{2}{9}=\frac{-2}{3}x-6\)
5. \(\frac{x}{2}+\frac{2}{7}=\frac{-2}{5}x-3\)
6. \(\frac{x}{5}-\frac{4}{9}=\frac{3}{2}x+6\)
7. \(\frac{x}{6}+\frac{5}{14}=\frac{-4}{5}x+2\)
8. \(\frac{x}{7}-\frac{4}{7}=\frac{1}{5}x+6\)
9. \(\frac{x}{3}+\frac{2}{9}=\frac{1}{4}x-6\)
10. \(\frac{x}{5}+\frac{2}{15}=\frac{7}{2}x-3\)
11. \(\frac{x}{2}+\frac{5}{6}=\frac{7}{3}x+8\)
12. \(\frac{x}{5}-\frac{3}{16}=\frac{5}{6}x-1\)

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{7}{4}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{231}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-24& = & 231x+792 \\\Leftrightarrow & 44x \color{red}{-24} \color{blue}{+24} \color{blue}{-231x} & = & \color{red}{231x} +792 \color{blue}{-231x} \color{blue}{+24} \\\Leftrightarrow & -187x& = & 816 \\\Leftrightarrow & \frac{-187x}{ \color{red}{-187} }& = & \frac{816}{-187} \\\Leftrightarrow & x = \frac{-48}{11} & & \\ & V = \left\{ \frac{-48}{11} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & 24x+240 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-24x} & = & \color{red}{24x} +240 \color{blue}{-24x} \color{blue}{-15} \\\Leftrightarrow & -8x& = & 225 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{225}{-8} \\\Leftrightarrow & x = \frac{-225}{8} & & \\ & V = \left\{ \frac{-225}{8} \right\} & \\\end{align}\)
3. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 105 }{ \color{blue}{385} })& = & (\frac{-154}{ \color{blue}{385} }x-\frac{1155}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-105& = & -154x-1155 \\\Leftrightarrow & 55x \color{red}{-105} \color{blue}{+105} \color{blue}{+154x} & = & \color{red}{-154x} -1155 \color{blue}{+154x} \color{blue}{+105} \\\Leftrightarrow & 209x& = & -1050 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-1050}{209} \\\Leftrightarrow & x = \frac{-1050}{209} & & \\ & V = \left\{ \frac{-1050}{209} \right\} & \\\end{align}\)
4. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }- \frac{ 14 }{ \color{blue}{63} })& = & (\frac{-42}{ \color{blue}{63} }x-\frac{378}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x-14& = & -42x-378 \\\Leftrightarrow & 9x \color{red}{-14} \color{blue}{+14} \color{blue}{+42x} & = & \color{red}{-42x} -378 \color{blue}{+42x} \color{blue}{+14} \\\Leftrightarrow & 51x& = & -364 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-364}{51} \\\Leftrightarrow & x = \frac{-364}{51} & & \\ & V = \left\{ \frac{-364}{51} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & -28x-210 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{+28x} & = & \color{red}{-28x} -210 \color{blue}{+28x} \color{blue}{-20} \\\Leftrightarrow & 63x& = & -230 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{-230}{63} \\\Leftrightarrow & x = \frac{-230}{63} & & \\ & V = \left\{ \frac{-230}{63} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{135}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 135x+540 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-135x} & = & \color{red}{135x} +540 \color{blue}{-135x} \color{blue}{+40} \\\Leftrightarrow & -117x& = & 580 \\\Leftrightarrow & \frac{-117x}{ \color{red}{-117} }& = & \frac{580}{-117} \\\Leftrightarrow & x = \frac{-580}{117} & & \\ & V = \left\{ \frac{-580}{117} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & -168x+420 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{+168x} & = & \color{red}{-168x} +420 \color{blue}{+168x} \color{blue}{-75} \\\Leftrightarrow & 203x& = & 345 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{345}{203} \\\Leftrightarrow & x = \frac{345}{203} & & \\ & V = \left\{ \frac{345}{203} \right\} & \\\end{align}\)
8. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 20 }{ \color{blue}{35} })& = & (\frac{7}{ \color{blue}{35} }x+\frac{210}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-20& = & 7x+210 \\\Leftrightarrow & 5x \color{red}{-20} \color{blue}{+20} \color{blue}{-7x} & = & \color{red}{7x} +210 \color{blue}{-7x} \color{blue}{+20} \\\Leftrightarrow & -2x& = & 230 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{230}{-2} \\\Leftrightarrow & x = -115 & & \\ & V = \left\{ -115 \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & 9x-216 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-9x} & = & \color{red}{9x} -216 \color{blue}{-9x} \color{blue}{-8} \\\Leftrightarrow & 3x& = & -224 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-224}{3} \\\Leftrightarrow & x = \frac{-224}{3} & & \\ & V = \left\{ \frac{-224}{3} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{105}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 105x-90 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-105x} & = & \color{red}{105x} -90 \color{blue}{-105x} \color{blue}{-4} \\\Leftrightarrow & -99x& = & -94 \\\Leftrightarrow & \frac{-99x}{ \color{red}{-99} }& = & \frac{-94}{-99} \\\Leftrightarrow & x = \frac{94}{99} & & \\ & V = \left\{ \frac{94}{99} \right\} & \\\end{align}\)
11. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{14}{ \color{blue}{6} }x+\frac{48}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 14x+48 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-14x} & = & \color{red}{14x} +48 \color{blue}{-14x} \color{blue}{-5} \\\Leftrightarrow & -11x& = & 43 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{43}{-11} \\\Leftrightarrow & x = \frac{-43}{11} & & \\ & V = \left\{ \frac{-43}{11} \right\} & \\\end{align}\)
12. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{5}{6}x-1 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{200}{ \color{blue}{240} }x-\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 200x-240 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-200x} & = & \color{red}{200x} -240 \color{blue}{-200x} \color{blue}{+45} \\\Leftrightarrow & -152x& = & -195 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{-195}{-152} \\\Leftrightarrow & x = \frac{195}{152} & & \\ & V = \left\{ \frac{195}{152} \right\} & \\\end{align}\)