Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & -294x-1050 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{+294x} & = & \color{red}{-294x} -1050 \color{blue}{+294x} \color{blue}{-45} \\\Leftrightarrow & 329x& = & -1095 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1095}{329} \\\Leftrightarrow & x = \frac{-1095}{329} & & \\ & V = \left\{ \frac{-1095}{329} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 9x-108 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-9x} & = & \color{red}{9x} -108 \color{blue}{-9x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & -116 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-116}{-3} \\\Leftrightarrow & x = \frac{116}{3} & & \\ & V = \left\{ \frac{116}{3} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 3, 6 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+25& = & 6x-180 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-6x} & = & \color{red}{6x} -180 \color{blue}{-6x} \color{blue}{-25} \\\Leftrightarrow & 4x& = & -205 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-205}{4} \\\Leftrightarrow & x = \frac{-205}{4} & & \\ & V = \left\{ \frac{-205}{4} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{3}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & 234x+1560 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{-234x} & = & \color{red}{234x} +1560 \color{blue}{-234x} \color{blue}{+90} \\\Leftrightarrow & -169x& = & 1650 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{1650}{-169} \\\Leftrightarrow & x = \frac{-1650}{169} & & \\ & V = \left\{ \frac{-1650}{169} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & -126x-630 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{+126x} & = & \color{red}{-126x} -630 \color{blue}{+126x} \color{blue}{+50} \\\Leftrightarrow & 141x& = & -580 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-580}{141} \\\Leftrightarrow & x = \frac{-580}{141} & & \\ & V = \left\{ \frac{-580}{141} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 196x-672 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-196x} & = & \color{red}{196x} -672 \color{blue}{-196x} \color{blue}{-24} \\\Leftrightarrow & -175x& = & -696 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-696}{-175} \\\Leftrightarrow & x = \frac{696}{175} & & \\ & V = \left\{ \frac{696}{175} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 35 }{ \color{blue}{84} })& = & (\frac{42}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+35& = & 42x-336 \\\Leftrightarrow & 12x \color{red}{+35} \color{blue}{-35} \color{blue}{-42x} & = & \color{red}{42x} -336 \color{blue}{-42x} \color{blue}{-35} \\\Leftrightarrow & -30x& = & -371 \\\Leftrightarrow & \frac{-30x}{ \color{red}{-30} }& = & \frac{-371}{-30} \\\Leftrightarrow & x = \frac{371}{30} & & \\ & V = \left\{ \frac{371}{30} \right\} & \\\end{align}\)
8. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 168 }{ \color{blue}{462} })& = & (\frac{77}{ \color{blue}{462} }x+\frac{2772}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-168& = & 77x+2772 \\\Leftrightarrow & 66x \color{red}{-168} \color{blue}{+168} \color{blue}{-77x} & = & \color{red}{77x} +2772 \color{blue}{-77x} \color{blue}{+168} \\\Leftrightarrow & -11x& = & 2940 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{2940}{-11} \\\Leftrightarrow & x = \frac{-2940}{11} & & \\ & V = \left\{ \frac{-2940}{11} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 35 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+35& = & -112x-42 \\\Leftrightarrow & 6x \color{red}{+35} \color{blue}{-35} \color{blue}{+112x} & = & \color{red}{-112x} -42 \color{blue}{+112x} \color{blue}{-35} \\\Leftrightarrow & 118x& = & -77 \\\Leftrightarrow & \frac{118x}{ \color{red}{118} }& = & \frac{-77}{118} \\\Leftrightarrow & x = \frac{-77}{118} & & \\ & V = \left\{ \frac{-77}{118} \right\} & \\\end{align}\)
10. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 80 }{ \color{blue}{220} })& = & (\frac{-176}{ \color{blue}{220} }x-\frac{660}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-80& = & -176x-660 \\\Leftrightarrow & 55x \color{red}{-80} \color{blue}{+80} \color{blue}{+176x} & = & \color{red}{-176x} -660 \color{blue}{+176x} \color{blue}{+80} \\\Leftrightarrow & 231x& = & -580 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-580}{231} \\\Leftrightarrow & x = \frac{-580}{231} & & \\ & V = \left\{ \frac{-580}{231} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & -260x-624 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{+260x} & = & \color{red}{-260x} -624 \color{blue}{+260x} \color{blue}{-36} \\\Leftrightarrow & 299x& = & -660 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-660}{299} \\\Leftrightarrow & x = \frac{-660}{299} & & \\ & V = \left\{ \frac{-660}{299} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-24& = & 7x+252 \\\Leftrightarrow & 6x \color{red}{-24} \color{blue}{+24} \color{blue}{-7x} & = & \color{red}{7x} +252 \color{blue}{-7x} \color{blue}{+24} \\\Leftrightarrow & -x& = & 276 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{276}{-1} \\\Leftrightarrow & x = -276 & & \\ & V = \left\{ -276 \right\} & \\\end{align}\)