Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{6}+\frac{2}{7}=\frac{-2}{5}x-6\)
2. \(\frac{x}{2}+\frac{2}{13}=\frac{5}{3}x-1\)
3. \(\frac{x}{5}+\frac{4}{11}=\frac{2}{3}x+1\)
4. \(\frac{x}{5}+\frac{4}{7}=\frac{-5}{3}x-6\)
5. \(\frac{x}{6}-\frac{3}{16}=\frac{1}{5}x-5\)
6. \(\frac{x}{6}+\frac{5}{6}=\frac{-4}{5}x+5\)
7. \(\frac{x}{7}-\frac{5}{7}=\frac{1}{2}x-6\)
8. \(\frac{x}{3}-\frac{2}{9}=\frac{-3}{4}x+7\)
9. \(\frac{x}{6}-\frac{4}{15}=\frac{-7}{5}x+4\)
10. \(\frac{x}{4}+\frac{3}{7}=\frac{4}{3}x-8\)
11. \(\frac{x}{4}+\frac{5}{14}=\frac{-2}{3}x+3\)
12. \(\frac{x}{2}-\frac{4}{15}=\frac{4}{3}x-5\)

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -84x-1260 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} -1260 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & -1320 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1320}{119} \\\Leftrightarrow & x = \frac{-1320}{119} & & \\ & V = \left\{ \frac{-1320}{119} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{130}{ \color{blue}{78} }x-\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 130x-78 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-130x} & = & \color{red}{130x} -78 \color{blue}{-130x} \color{blue}{-12} \\\Leftrightarrow & -91x& = & -90 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-90}{-91} \\\Leftrightarrow & x = \frac{90}{91} & & \\ & V = \left\{ \frac{90}{91} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{2}{3}x+1 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 60 }{ \color{blue}{165} })& = & (\frac{110}{ \color{blue}{165} }x+\frac{165}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+60& = & 110x+165 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-110x} & = & \color{red}{110x} +165 \color{blue}{-110x} \color{blue}{-60} \\\Leftrightarrow & -77x& = & 105 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{105}{-77} \\\Leftrightarrow & x = \frac{-15}{11} & & \\ & V = \left\{ \frac{-15}{11} \right\} & \\\end{align}\)
4. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+60& = & -175x-630 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{+175x} & = & \color{red}{-175x} -630 \color{blue}{+175x} \color{blue}{-60} \\\Leftrightarrow & 196x& = & -690 \\\Leftrightarrow & \frac{196x}{ \color{red}{196} }& = & \frac{-690}{196} \\\Leftrightarrow & x = \frac{-345}{98} & & \\ & V = \left\{ \frac{-345}{98} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 48x-1200 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-48x} & = & \color{red}{48x} -1200 \color{blue}{-48x} \color{blue}{+45} \\\Leftrightarrow & -8x& = & -1155 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-1155}{-8} \\\Leftrightarrow & x = \frac{1155}{8} & & \\ & V = \left\{ \frac{1155}{8} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -24x+150 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} +150 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 29x& = & 125 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{125}{29} \\\Leftrightarrow & x = \frac{125}{29} & & \\ & V = \left\{ \frac{125}{29} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 10 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-10& = & 7x-84 \\\Leftrightarrow & 2x \color{red}{-10} \color{blue}{+10} \color{blue}{-7x} & = & \color{red}{7x} -84 \color{blue}{-7x} \color{blue}{+10} \\\Leftrightarrow & -5x& = & -74 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-74}{-5} \\\Leftrightarrow & x = \frac{74}{5} & & \\ & V = \left\{ \frac{74}{5} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & -27x+252 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{+27x} & = & \color{red}{-27x} +252 \color{blue}{+27x} \color{blue}{+8} \\\Leftrightarrow & 39x& = & 260 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{260}{39} \\\Leftrightarrow & x = \frac{20}{3} & & \\ & V = \left\{ \frac{20}{3} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -42x+120 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} +120 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 47x& = & 128 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{128}{47} \\\Leftrightarrow & x = \frac{128}{47} & & \\ & V = \left\{ \frac{128}{47} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 112x-672 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-112x} & = & \color{red}{112x} -672 \color{blue}{-112x} \color{blue}{-36} \\\Leftrightarrow & -91x& = & -708 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-708}{-91} \\\Leftrightarrow & x = \frac{708}{91} & & \\ & V = \left\{ \frac{708}{91} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{14}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 30 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+30& = & -56x+252 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{+56x} & = & \color{red}{-56x} +252 \color{blue}{+56x} \color{blue}{-30} \\\Leftrightarrow & 77x& = & 222 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{222}{77} \\\Leftrightarrow & x = \frac{222}{77} & & \\ & V = \left\{ \frac{222}{77} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{40}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & 40x-150 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-40x} & = & \color{red}{40x} -150 \color{blue}{-40x} \color{blue}{+8} \\\Leftrightarrow & -25x& = & -142 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-142}{-25} \\\Leftrightarrow & x = \frac{142}{25} & & \\ & V = \left\{ \frac{142}{25} \right\} & \\\end{align}\)