Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{4}x-7 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{12}{ \color{blue}{48} }x-\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 12x-336 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} -336 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & 4x& = & -327 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-327}{4} \\\Leftrightarrow & x = \frac{-327}{4} & & \\ & V = \left\{ \frac{-327}{4} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 126x+1470 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-126x} & = & \color{red}{126x} +1470 \color{blue}{-126x} \color{blue}{-120} \\\Leftrightarrow & -91x& = & 1350 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{1350}{-91} \\\Leftrightarrow & x = \frac{-1350}{91} & & \\ & V = \left\{ \frac{-1350}{91} \right\} & \\\end{align}\)
3. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 70 }{ \color{blue}{455} })& = & (\frac{91}{ \color{blue}{455} }x-\frac{1365}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+70& = & 91x-1365 \\\Leftrightarrow & 65x \color{red}{+70} \color{blue}{-70} \color{blue}{-91x} & = & \color{red}{91x} -1365 \color{blue}{-91x} \color{blue}{-70} \\\Leftrightarrow & -26x& = & -1435 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-1435}{-26} \\\Leftrightarrow & x = \frac{1435}{26} & & \\ & V = \left\{ \frac{1435}{26} \right\} & \\\end{align}\)
4. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-147}{ \color{blue}{105} }x-\frac{840}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-30& = & -147x-840 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{+147x} & = & \color{red}{-147x} -840 \color{blue}{+147x} \color{blue}{+30} \\\Leftrightarrow & 182x& = & -810 \\\Leftrightarrow & \frac{182x}{ \color{red}{182} }& = & \frac{-810}{182} \\\Leftrightarrow & x = \frac{-405}{91} & & \\ & V = \left\{ \frac{-405}{91} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & -56x-140 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+56x} & = & \color{red}{-56x} -140 \color{blue}{+56x} \color{blue}{+40} \\\Leftrightarrow & 91x& = & -100 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-100}{91} \\\Leftrightarrow & x = \frac{-100}{91} & & \\ & V = \left\{ \frac{-100}{91} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{30}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 30x+360 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-30x} & = & \color{red}{30x} +360 \color{blue}{-30x} \color{blue}{+25} \\\Leftrightarrow & -18x& = & 385 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{385}{-18} \\\Leftrightarrow & x = \frac{-385}{18} & & \\ & V = \left\{ \frac{-385}{18} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 252x+840 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-252x} & = & \color{red}{252x} +840 \color{blue}{-252x} \color{blue}{-60} \\\Leftrightarrow & -217x& = & 780 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{780}{-217} \\\Leftrightarrow & x = \frac{-780}{217} & & \\ & V = \left\{ \frac{-780}{217} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-8}{3}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & -208x-390 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+208x} & = & \color{red}{-208x} -390 \color{blue}{+208x} \color{blue}{+24} \\\Leftrightarrow & 247x& = & -366 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-366}{247} \\\Leftrightarrow & x = \frac{-366}{247} & & \\ & V = \left\{ \frac{-366}{247} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{3}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{84}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & 84x-280 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{-84x} & = & \color{red}{84x} -280 \color{blue}{-84x} \color{blue}{+80} \\\Leftrightarrow & -49x& = & -200 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-200}{-49} \\\Leftrightarrow & x = \frac{200}{49} & & \\ & V = \left\{ \frac{200}{49} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 35x-210 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{+60} \\\Leftrightarrow & 7x& = & -150 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-150}{7} \\\Leftrightarrow & x = \frac{-150}{7} & & \\ & V = \left\{ \frac{-150}{7} \right\} & \\\end{align}\)
11. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-135}{ \color{blue}{180} }x+\frac{540}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x+80& = & -135x+540 \\\Leftrightarrow & 36x \color{red}{+80} \color{blue}{-80} \color{blue}{+135x} & = & \color{red}{-135x} +540 \color{blue}{+135x} \color{blue}{-80} \\\Leftrightarrow & 171x& = & 460 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{460}{171} \\\Leftrightarrow & x = \frac{460}{171} & & \\ & V = \left\{ \frac{460}{171} \right\} & \\\end{align}\)
12. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{2}{ \color{blue}{6} }x+\frac{48}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x-5& = & 2x+48 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-2x} & = & \color{red}{2x} +48 \color{blue}{-2x} \color{blue}{+5} \\\Leftrightarrow & x& = & 53 \\ & V = \left\{ 53 \right\} & \\\end{align}\)