Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}-\frac{5}{11}=\frac{1}{2}x+6\)
2. \(\frac{x}{3}+\frac{3}{16}=\frac{1}{2}x-2\)
3. \(\frac{x}{7}-\frac{3}{11}=\frac{1}{2}x+3\)
4. \(\frac{x}{5}+\frac{2}{13}=\frac{7}{6}x+5\)
5. \(\frac{x}{6}-\frac{4}{7}=\frac{2}{5}x+5\)
6. \(\frac{x}{5}+\frac{2}{9}=\frac{-3}{4}x-8\)
7. \(\frac{x}{3}+\frac{2}{7}=\frac{1}{2}x-6\)
8. \(\frac{x}{2}-\frac{2}{7}=\frac{5}{3}x-3\)
9. \(\frac{x}{2}-\frac{4}{7}=\frac{1}{5}x+6\)
10. \(\frac{x}{7}-\frac{5}{14}=\frac{2}{5}x+8\)
11. \(\frac{x}{3}-\frac{4}{9}=\frac{-3}{4}x-3\)
12. \(\frac{x}{3}-\frac{3}{10}=\frac{1}{2}x-6\)

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

Verbetersleutel

1. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 55x+660 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-55x} & = & \color{red}{55x} +660 \color{blue}{-55x} \color{blue}{+50} \\\Leftrightarrow & -33x& = & 710 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{710}{-33} \\\Leftrightarrow & x = \frac{-710}{33} & & \\ & V = \left\{ \frac{-710}{33} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x-96 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} -96 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -105 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-105}{-8} \\\Leftrightarrow & x = \frac{105}{8} & & \\ & V = \left\{ \frac{105}{8} \right\} & \\\end{align}\)
3. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 42 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{462}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-42& = & 77x+462 \\\Leftrightarrow & 22x \color{red}{-42} \color{blue}{+42} \color{blue}{-77x} & = & \color{red}{77x} +462 \color{blue}{-77x} \color{blue}{+42} \\\Leftrightarrow & -55x& = & 504 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{504}{-55} \\\Leftrightarrow & x = \frac{-504}{55} & & \\ & V = \left\{ \frac{-504}{55} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{7}{6}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{455}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 455x+1950 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-455x} & = & \color{red}{455x} +1950 \color{blue}{-455x} \color{blue}{-60} \\\Leftrightarrow & -377x& = & 1890 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{1890}{-377} \\\Leftrightarrow & x = \frac{-1890}{377} & & \\ & V = \left\{ \frac{-1890}{377} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{2}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 84x+1050 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-84x} & = & \color{red}{84x} +1050 \color{blue}{-84x} \color{blue}{+120} \\\Leftrightarrow & -49x& = & 1170 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{1170}{-49} \\\Leftrightarrow & x = \frac{-1170}{49} & & \\ & V = \left\{ \frac{-1170}{49} \right\} & \\\end{align}\)
6. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{-3}{4}x-8 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-135}{ \color{blue}{180} }x-\frac{1440}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x+40& = & -135x-1440 \\\Leftrightarrow & 36x \color{red}{+40} \color{blue}{-40} \color{blue}{+135x} & = & \color{red}{-135x} -1440 \color{blue}{+135x} \color{blue}{-40} \\\Leftrightarrow & 171x& = & -1480 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{-1480}{171} \\\Leftrightarrow & x = \frac{-1480}{171} & & \\ & V = \left\{ \frac{-1480}{171} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & -264 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-264}{-7} \\\Leftrightarrow & x = \frac{264}{7} & & \\ & V = \left\{ \frac{264}{7} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{5}{3}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{70}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 70x-126 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-70x} & = & \color{red}{70x} -126 \color{blue}{-70x} \color{blue}{+12} \\\Leftrightarrow & -49x& = & -114 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-114}{-49} \\\Leftrightarrow & x = \frac{114}{49} & & \\ & V = \left\{ \frac{114}{49} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & 14x+420 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-14x} & = & \color{red}{14x} +420 \color{blue}{-14x} \color{blue}{+40} \\\Leftrightarrow & 21x& = & 460 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{460}{21} \\\Leftrightarrow & x = \frac{460}{21} & & \\ & V = \left\{ \frac{460}{21} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{14}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 25 }{ \color{blue}{70} })& = & (\frac{28}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-25& = & 28x+560 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-28x} & = & \color{red}{28x} +560 \color{blue}{-28x} \color{blue}{+25} \\\Leftrightarrow & -18x& = & 585 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{585}{-18} \\\Leftrightarrow & x = \frac{-65}{2} & & \\ & V = \left\{ \frac{-65}{2} \right\} & \\\end{align}\)
11. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x-\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-16& = & -27x-108 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{+27x} & = & \color{red}{-27x} -108 \color{blue}{+27x} \color{blue}{+16} \\\Leftrightarrow & 39x& = & -92 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-92}{39} \\\Leftrightarrow & x = \frac{-92}{39} & & \\ & V = \left\{ \frac{-92}{39} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 15x-180 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-15x} & = & \color{red}{15x} -180 \color{blue}{-15x} \color{blue}{+9} \\\Leftrightarrow & -5x& = & -171 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-171}{-5} \\\Leftrightarrow & x = \frac{171}{5} & & \\ & V = \left\{ \frac{171}{5} \right\} & \\\end{align}\)