Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{195}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+24& = & 195x+624 \\\Leftrightarrow & 52x \color{red}{+24} \color{blue}{-24} \color{blue}{-195x} & = & \color{red}{195x} +624 \color{blue}{-195x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & 600 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{600}{-143} \\\Leftrightarrow & x = \frac{-600}{143} & & \\ & V = \left\{ \frac{-600}{143} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -84x+420 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+84x} & = & \color{red}{-84x} +420 \color{blue}{+84x} \color{blue}{-120} \\\Leftrightarrow & 119x& = & 300 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{300}{119} \\\Leftrightarrow & x = \frac{300}{119} & & \\ & V = \left\{ \frac{300}{119} \right\} & \\\end{align}\)
3. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{1}{4}x+1 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{28}{ \color{blue}{112} }x+\frac{112}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & 28x+112 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{-28x} & = & \color{red}{28x} +112 \color{blue}{-28x} \color{blue}{+35} \\\Leftrightarrow & -12x& = & 147 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{147}{-12} \\\Leftrightarrow & x = \frac{-49}{4} & & \\ & V = \left\{ \frac{-49}{4} \right\} & \\\end{align}\)
4. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{8}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x+\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-9& = & 56x+120 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-56x} & = & \color{red}{56x} +120 \color{blue}{-56x} \color{blue}{+9} \\\Leftrightarrow & -50x& = & 129 \\\Leftrightarrow & \frac{-50x}{ \color{red}{-50} }& = & \frac{129}{-50} \\\Leftrightarrow & x = \frac{-129}{50} & & \\ & V = \left\{ \frac{-129}{50} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & 6x+120 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{-6x} & = & \color{red}{6x} +120 \color{blue}{-6x} \color{blue}{+9} \\\Leftrightarrow & -x& = & 129 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{129}{-1} \\\Leftrightarrow & x = -129 & & \\ & V = \left\{ -129 \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 28 }{ \color{blue}{210} })& = & (\frac{315}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-28& = & 315x-1260 \\\Leftrightarrow & 30x \color{red}{-28} \color{blue}{+28} \color{blue}{-315x} & = & \color{red}{315x} -1260 \color{blue}{-315x} \color{blue}{+28} \\\Leftrightarrow & -285x& = & -1232 \\\Leftrightarrow & \frac{-285x}{ \color{red}{-285} }& = & \frac{-1232}{-285} \\\Leftrightarrow & x = \frac{1232}{285} & & \\ & V = \left\{ \frac{1232}{285} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & -40x+120 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{+40x} & = & \color{red}{-40x} +120 \color{blue}{+40x} \color{blue}{+16} \\\Leftrightarrow & 55x& = & 136 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{136}{55} \\\Leftrightarrow & x = \frac{136}{55} & & \\ & V = \left\{ \frac{136}{55} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{11}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 36 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+36& = & -220x+396 \\\Leftrightarrow & 33x \color{red}{+36} \color{blue}{-36} \color{blue}{+220x} & = & \color{red}{-220x} +396 \color{blue}{+220x} \color{blue}{-36} \\\Leftrightarrow & 253x& = & 360 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{360}{253} \\\Leftrightarrow & x = \frac{360}{253} & & \\ & V = \left\{ \frac{360}{253} \right\} & \\\end{align}\)
9. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }+ \frac{ 28 }{ \color{blue}{63} })& = & (\frac{21}{ \color{blue}{63} }x+\frac{252}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x+28& = & 21x+252 \\\Leftrightarrow & 9x \color{red}{+28} \color{blue}{-28} \color{blue}{-21x} & = & \color{red}{21x} +252 \color{blue}{-21x} \color{blue}{-28} \\\Leftrightarrow & -12x& = & 224 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{224}{-12} \\\Leftrightarrow & x = \frac{-56}{3} & & \\ & V = \left\{ \frac{-56}{3} \right\} & \\\end{align}\)
10. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 288x+1920 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-288x} & = & \color{red}{288x} +1920 \color{blue}{-288x} \color{blue}{+45} \\\Leftrightarrow & -248x& = & 1965 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{1965}{-248} \\\Leftrightarrow & x = \frac{-1965}{248} & & \\ & V = \left\{ \frac{-1965}{248} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{4}{5}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & 312x+1950 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{-312x} & = & \color{red}{312x} +1950 \color{blue}{-312x} \color{blue}{-120} \\\Leftrightarrow & -247x& = & 1830 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{1830}{-247} \\\Leftrightarrow & x = \frac{-1830}{247} & & \\ & V = \left\{ \frac{-1830}{247} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{-88}{ \color{blue}{110} }x+\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+50& = & -88x+440 \\\Leftrightarrow & 55x \color{red}{+50} \color{blue}{-50} \color{blue}{+88x} & = & \color{red}{-88x} +440 \color{blue}{+88x} \color{blue}{-50} \\\Leftrightarrow & 143x& = & 390 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{390}{143} \\\Leftrightarrow & x = \frac{30}{11} & & \\ & V = \left\{ \frac{30}{11} \right\} & \\\end{align}\)