Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{45}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 45x-126 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-45x} & = & \color{red}{45x} -126 \color{blue}{-45x} \color{blue}{-8} \\\Leftrightarrow & -39x& = & -134 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-134}{-39} \\\Leftrightarrow & x = \frac{134}{39} & & \\ & V = \left\{ \frac{134}{39} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & 6x+60 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-6x} & = & \color{red}{6x} +60 \color{blue}{-6x} \color{blue}{+25} \\\Leftrightarrow & -x& = & 85 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{85}{-1} \\\Leftrightarrow & x = -85 & & \\ & V = \left\{ -85 \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & -12x+150 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} +150 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 17x& = & 146 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{146}{17} \\\Leftrightarrow & x = \frac{146}{17} & & \\ & V = \left\{ \frac{146}{17} \right\} & \\\end{align}\)
4. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x+\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & -64x+240 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{+64x} & = & \color{red}{-64x} +240 \color{blue}{+64x} \color{blue}{+25} \\\Leftrightarrow & 84x& = & 265 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{265}{84} \\\Leftrightarrow & x = \frac{265}{84} & & \\ & V = \left\{ \frac{265}{84} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 196x-504 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-196x} & = & \color{red}{196x} -504 \color{blue}{-196x} \color{blue}{+24} \\\Leftrightarrow & -175x& = & -480 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-480}{-175} \\\Leftrightarrow & x = \frac{96}{35} & & \\ & V = \left\{ \frac{96}{35} \right\} & \\\end{align}\)
6. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{2}{5}x+4 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{104}{ \color{blue}{260} }x+\frac{1040}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-60& = & 104x+1040 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-104x} & = & \color{red}{104x} +1040 \color{blue}{-104x} \color{blue}{+60} \\\Leftrightarrow & -39x& = & 1100 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{1100}{-39} \\\Leftrightarrow & x = \frac{-1100}{39} & & \\ & V = \left\{ \frac{-1100}{39} \right\} & \\\end{align}\)
7. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 70 }{ \color{blue}{154} })& = & (\frac{385}{ \color{blue}{154} }x+\frac{462}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+70& = & 385x+462 \\\Leftrightarrow & 22x \color{red}{+70} \color{blue}{-70} \color{blue}{-385x} & = & \color{red}{385x} +462 \color{blue}{-385x} \color{blue}{-70} \\\Leftrightarrow & -363x& = & 392 \\\Leftrightarrow & \frac{-363x}{ \color{red}{-363} }& = & \frac{392}{-363} \\\Leftrightarrow & x = \frac{-392}{363} & & \\ & V = \left\{ \frac{-392}{363} \right\} & \\\end{align}\)
8. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }+ \frac{ 20 }{ \color{blue}{45} })& = & (\frac{105}{ \color{blue}{45} }x+\frac{90}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x+20& = & 105x+90 \\\Leftrightarrow & 9x \color{red}{+20} \color{blue}{-20} \color{blue}{-105x} & = & \color{red}{105x} +90 \color{blue}{-105x} \color{blue}{-20} \\\Leftrightarrow & -96x& = & 70 \\\Leftrightarrow & \frac{-96x}{ \color{red}{-96} }& = & \frac{70}{-96} \\\Leftrightarrow & x = \frac{-35}{48} & & \\ & V = \left\{ \frac{-35}{48} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x+\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & -48x+54 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{+48x} & = & \color{red}{-48x} +54 \color{blue}{+48x} \color{blue}{-10} \\\Leftrightarrow & 57x& = & 44 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{44}{57} \\\Leftrightarrow & x = \frac{44}{57} & & \\ & V = \left\{ \frac{44}{57} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & -40x+420 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{+40x} & = & \color{red}{-40x} +420 \color{blue}{+40x} \color{blue}{-18} \\\Leftrightarrow & 55x& = & 402 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{402}{55} \\\Leftrightarrow & x = \frac{402}{55} & & \\ & V = \left\{ \frac{402}{55} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-8}{3}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & -208x-390 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+208x} & = & \color{red}{-208x} -390 \color{blue}{+208x} \color{blue}{-24} \\\Leftrightarrow & 247x& = & -414 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-414}{247} \\\Leftrightarrow & x = \frac{-414}{247} & & \\ & V = \left\{ \frac{-414}{247} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-80}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -80x-60 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+80x} & = & \color{red}{-80x} -60 \color{blue}{+80x} \color{blue}{+8} \\\Leftrightarrow & 95x& = & -52 \\\Leftrightarrow & \frac{95x}{ \color{red}{95} }& = & \frac{-52}{95} \\\Leftrightarrow & x = \frac{-52}{95} & & \\ & V = \left\{ \frac{-52}{95} \right\} & \\\end{align}\)