Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{1}{4}x+8 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x+\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & 9x+288 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} +288 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & 3x& = & 296 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{296}{3} \\\Leftrightarrow & x = \frac{296}{3} & & \\ & V = \left\{ \frac{296}{3} \right\} & \\\end{align}\)
2. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{8}& = & \frac{3}{5}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{72}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-45& = & 72x-360 \\\Leftrightarrow & 20x \color{red}{-45} \color{blue}{+45} \color{blue}{-72x} & = & \color{red}{72x} -360 \color{blue}{-72x} \color{blue}{+45} \\\Leftrightarrow & -52x& = & -315 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{-315}{-52} \\\Leftrightarrow & x = \frac{315}{52} & & \\ & V = \left\{ \frac{315}{52} \right\} & \\\end{align}\)
3. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & 396x-660 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-396x} & = & \color{red}{396x} -660 \color{blue}{-396x} \color{blue}{-60} \\\Leftrightarrow & -341x& = & -720 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-720}{-341} \\\Leftrightarrow & x = \frac{720}{341} & & \\ & V = \left\{ \frac{720}{341} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x-\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 88x-198 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-88x} & = & \color{red}{88x} -198 \color{blue}{-88x} \color{blue}{+12} \\\Leftrightarrow & -55x& = & -186 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-186}{-55} \\\Leftrightarrow & x = \frac{186}{55} & & \\ & V = \left\{ \frac{186}{55} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & -126x-450 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{+126x} & = & \color{red}{-126x} -450 \color{blue}{+126x} \color{blue}{+50} \\\Leftrightarrow & 141x& = & -400 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-400}{141} \\\Leftrightarrow & x = \frac{-400}{141} & & \\ & V = \left\{ \frac{-400}{141} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x-\frac{12}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x+5& = & -32x-12 \\\Leftrightarrow & 6x \color{red}{+5} \color{blue}{-5} \color{blue}{+32x} & = & \color{red}{-32x} -12 \color{blue}{+32x} \color{blue}{-5} \\\Leftrightarrow & 38x& = & -17 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{-17}{38} \\\Leftrightarrow & x = \frac{-17}{38} & & \\ & V = \left\{ \frac{-17}{38} \right\} & \\\end{align}\)
7. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 80 }{ \color{blue}{220} })& = & (\frac{-176}{ \color{blue}{220} }x+\frac{440}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+80& = & -176x+440 \\\Leftrightarrow & 55x \color{red}{+80} \color{blue}{-80} \color{blue}{+176x} & = & \color{red}{-176x} +440 \color{blue}{+176x} \color{blue}{-80} \\\Leftrightarrow & 231x& = & 360 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{360}{231} \\\Leftrightarrow & x = \frac{120}{77} & & \\ & V = \left\{ \frac{120}{77} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{4}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{20}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 20x-400 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-20x} & = & \color{red}{20x} -400 \color{blue}{-20x} \color{blue}{+25} \\\Leftrightarrow & -4x& = & -375 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-375}{-4} \\\Leftrightarrow & x = \frac{375}{4} & & \\ & V = \left\{ \frac{375}{4} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -84x-1050 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+84x} & = & \color{red}{-84x} -1050 \color{blue}{+84x} \color{blue}{+90} \\\Leftrightarrow & 119x& = & -960 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-960}{119} \\\Leftrightarrow & x = \frac{-960}{119} & & \\ & V = \left\{ \frac{-960}{119} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & -48x+420 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{+48x} & = & \color{red}{-48x} +420 \color{blue}{+48x} \color{blue}{-25} \\\Leftrightarrow & 58x& = & 395 \\\Leftrightarrow & \frac{58x}{ \color{red}{58} }& = & \frac{395}{58} \\\Leftrightarrow & x = \frac{395}{58} & & \\ & V = \left\{ \frac{395}{58} \right\} & \\\end{align}\)
11. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{-88}{ \color{blue}{110} }x+\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-50& = & -88x+880 \\\Leftrightarrow & 55x \color{red}{-50} \color{blue}{+50} \color{blue}{+88x} & = & \color{red}{-88x} +880 \color{blue}{+88x} \color{blue}{+50} \\\Leftrightarrow & 143x& = & 930 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{930}{143} \\\Leftrightarrow & x = \frac{930}{143} & & \\ & V = \left\{ \frac{930}{143} \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & -110x+528 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+110x} & = & \color{red}{-110x} +528 \color{blue}{+110x} \color{blue}{-24} \\\Leftrightarrow & 143x& = & 504 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{504}{143} \\\Leftrightarrow & x = \frac{504}{143} & & \\ & V = \left\{ \frac{504}{143} \right\} & \\\end{align}\)