Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+15& = & 64x+240 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{-64x} & = & \color{red}{64x} +240 \color{blue}{-64x} \color{blue}{-15} \\\Leftrightarrow & -52x& = & 225 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{225}{-52} \\\Leftrightarrow & x = \frac{-225}{52} & & \\ & V = \left\{ \frac{-225}{52} \right\} & \\\end{align}\)
2. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 105 }{ \color{blue}{336} })& = & (\frac{112}{ \color{blue}{336} }x-\frac{1344}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+105& = & 112x-1344 \\\Leftrightarrow & 48x \color{red}{+105} \color{blue}{-105} \color{blue}{-112x} & = & \color{red}{112x} -1344 \color{blue}{-112x} \color{blue}{-105} \\\Leftrightarrow & -64x& = & -1449 \\\Leftrightarrow & \frac{-64x}{ \color{red}{-64} }& = & \frac{-1449}{-64} \\\Leftrightarrow & x = \frac{1449}{64} & & \\ & V = \left\{ \frac{1449}{64} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{45}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-25& = & 45x-60 \\\Leftrightarrow & 6x \color{red}{-25} \color{blue}{+25} \color{blue}{-45x} & = & \color{red}{45x} -60 \color{blue}{-45x} \color{blue}{+25} \\\Leftrightarrow & -39x& = & -35 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-35}{-39} \\\Leftrightarrow & x = \frac{35}{39} & & \\ & V = \left\{ \frac{35}{39} \right\} & \\\end{align}\)
4. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{5}{2}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{200}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 200x+400 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-200x} & = & \color{red}{200x} +400 \color{blue}{-200x} \color{blue}{-25} \\\Leftrightarrow & -184x& = & 375 \\\Leftrightarrow & \frac{-184x}{ \color{red}{-184} }& = & \frac{375}{-184} \\\Leftrightarrow & x = \frac{-375}{184} & & \\ & V = \left\{ \frac{-375}{184} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{2}{5}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 28 }{ \color{blue}{105} })& = & (\frac{42}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-28& = & 42x-630 \\\Leftrightarrow & 15x \color{red}{-28} \color{blue}{+28} \color{blue}{-42x} & = & \color{red}{42x} -630 \color{blue}{-42x} \color{blue}{+28} \\\Leftrightarrow & -27x& = & -602 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-602}{-27} \\\Leftrightarrow & x = \frac{602}{27} & & \\ & V = \left\{ \frac{602}{27} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{1}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 28x-672 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-28x} & = & \color{red}{28x} -672 \color{blue}{-28x} \color{blue}{+36} \\\Leftrightarrow & -7x& = & -636 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-636}{-7} \\\Leftrightarrow & x = \frac{636}{7} & & \\ & V = \left\{ \frac{636}{7} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -56x-84 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+56x} & = & \color{red}{-56x} -84 \color{blue}{+56x} \color{blue}{+48} \\\Leftrightarrow & 77x& = & -36 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-36}{77} \\\Leftrightarrow & x = \frac{-36}{77} & & \\ & V = \left\{ \frac{-36}{77} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{5}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 140x+672 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-140x} & = & \color{red}{140x} +672 \color{blue}{-140x} \color{blue}{+24} \\\Leftrightarrow & -119x& = & 696 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{696}{-119} \\\Leftrightarrow & x = \frac{-696}{119} & & \\ & V = \left\{ \frac{-696}{119} \right\} & \\\end{align}\)
9. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-135}{ \color{blue}{180} }x+\frac{540}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x+80& = & -135x+540 \\\Leftrightarrow & 36x \color{red}{+80} \color{blue}{-80} \color{blue}{+135x} & = & \color{red}{-135x} +540 \color{blue}{+135x} \color{blue}{-80} \\\Leftrightarrow & 171x& = & 460 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{460}{171} \\\Leftrightarrow & x = \frac{460}{171} & & \\ & V = \left\{ \frac{460}{171} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{3}{4}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{45}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-8& = & 45x+480 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-45x} & = & \color{red}{45x} +480 \color{blue}{-45x} \color{blue}{+8} \\\Leftrightarrow & -33x& = & 488 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{488}{-33} \\\Leftrightarrow & x = \frac{-488}{33} & & \\ & V = \left\{ \frac{-488}{33} \right\} & \\\end{align}\)
11. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{2}{3}x-6 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{4}{ \color{blue}{6} }x-\frac{36}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 4x-36 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-4x} & = & \color{red}{4x} -36 \color{blue}{-4x} \color{blue}{-5} \\\Leftrightarrow & -x& = & -41 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-41}{-1} \\\Leftrightarrow & x = 41 & & \\ & V = \left\{ 41 \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{2}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 84x-1470 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-84x} & = & \color{red}{84x} -1470 \color{blue}{-84x} \color{blue}{+90} \\\Leftrightarrow & -49x& = & -1380 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-1380}{-49} \\\Leftrightarrow & x = \frac{1380}{49} & & \\ & V = \left\{ \frac{1380}{49} \right\} & \\\end{align}\)