Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{260}{ \color{blue}{195} }x-\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+30& = & 260x-390 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{-260x} & = & \color{red}{260x} -390 \color{blue}{-260x} \color{blue}{-30} \\\Leftrightarrow & -221x& = & -420 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{-420}{-221} \\\Leftrightarrow & x = \frac{420}{221} & & \\ & V = \left\{ \frac{420}{221} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 56x+168 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-56x} & = & \color{red}{56x} +168 \color{blue}{-56x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & 192 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{192}{-35} \\\Leftrightarrow & x = \frac{-192}{35} & & \\ & V = \left\{ \frac{-192}{35} \right\} & \\\end{align}\)
3. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{1274}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+42& = & 91x-1274 \\\Leftrightarrow & 26x \color{red}{+42} \color{blue}{-42} \color{blue}{-91x} & = & \color{red}{91x} -1274 \color{blue}{-91x} \color{blue}{-42} \\\Leftrightarrow & -65x& = & -1316 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-1316}{-65} \\\Leftrightarrow & x = \frac{1316}{65} & & \\ & V = \left\{ \frac{1316}{65} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & 88x+528 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-88x} & = & \color{red}{88x} +528 \color{blue}{-88x} \color{blue}{-24} \\\Leftrightarrow & -55x& = & 504 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{504}{-55} \\\Leftrightarrow & x = \frac{-504}{55} & & \\ & V = \left\{ \frac{-504}{55} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 3, 12 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-25& = & 12x+180 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{-12x} & = & \color{red}{12x} +180 \color{blue}{-12x} \color{blue}{+25} \\\Leftrightarrow & 8x& = & 205 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{205}{8} \\\Leftrightarrow & x = \frac{205}{8} & & \\ & V = \left\{ \frac{205}{8} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 28x-504 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-28x} & = & \color{red}{28x} -504 \color{blue}{-28x} \color{blue}{+36} \\\Leftrightarrow & -7x& = & -468 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-468}{-7} \\\Leftrightarrow & x = \frac{468}{7} & & \\ & V = \left\{ \frac{468}{7} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{11}& = & \frac{2}{3}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 36 }{ \color{blue}{132} })& = & (\frac{88}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+36& = & 88x+792 \\\Leftrightarrow & 33x \color{red}{+36} \color{blue}{-36} \color{blue}{-88x} & = & \color{red}{88x} +792 \color{blue}{-88x} \color{blue}{-36} \\\Leftrightarrow & -55x& = & 756 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{756}{-55} \\\Leftrightarrow & x = \frac{-756}{55} & & \\ & V = \left\{ \frac{-756}{55} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{7}{4}x-8 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{63}{ \color{blue}{36} }x-\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & 63x-288 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-63x} & = & \color{red}{63x} -288 \color{blue}{-63x} \color{blue}{+8} \\\Leftrightarrow & -51x& = & -280 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{-280}{-51} \\\Leftrightarrow & x = \frac{280}{51} & & \\ & V = \left\{ \frac{280}{51} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{5}{6}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 25x-180 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-25x} & = & \color{red}{25x} -180 \color{blue}{-25x} \color{blue}{+8} \\\Leftrightarrow & -19x& = & -172 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-172}{-19} \\\Leftrightarrow & x = \frac{172}{19} & & \\ & V = \left\{ \frac{172}{19} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 78x-2730 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-78x} & = & \color{red}{78x} -2730 \color{blue}{-78x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & -2670 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-2670}{-13} \\\Leftrightarrow & x = \frac{2670}{13} & & \\ & V = \left\{ \frac{2670}{13} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{6}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+90& = & 65x+780 \\\Leftrightarrow & 78x \color{red}{+90} \color{blue}{-90} \color{blue}{-65x} & = & \color{red}{65x} +780 \color{blue}{-65x} \color{blue}{-90} \\\Leftrightarrow & 13x& = & 690 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{690}{13} \\\Leftrightarrow & x = \frac{690}{13} & & \\ & V = \left\{ \frac{690}{13} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 112x+672 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-112x} & = & \color{red}{112x} +672 \color{blue}{-112x} \color{blue}{+48} \\\Leftrightarrow & -91x& = & 720 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{720}{-91} \\\Leftrightarrow & x = \frac{-720}{91} & & \\ & V = \left\{ \frac{-720}{91} \right\} & \\\end{align}\)