Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -312x-1560 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+312x} & = & \color{red}{-312x} -1560 \color{blue}{+312x} \color{blue}{-90} \\\Leftrightarrow & 377x& = & -1650 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-1650}{377} \\\Leftrightarrow & x = \frac{-1650}{377} & & \\ & V = \left\{ \frac{-1650}{377} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 9x-108 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} -108 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & -100 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-100}{-3} \\\Leftrightarrow & x = \frac{100}{3} & & \\ & V = \left\{ \frac{100}{3} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x+210 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & 190 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{190}{-21} \\\Leftrightarrow & x = \frac{-190}{21} & & \\ & V = \left\{ \frac{-190}{21} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{-5}{6}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-9& = & -35x+210 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{+35x} & = & \color{red}{-35x} +210 \color{blue}{+35x} \color{blue}{+9} \\\Leftrightarrow & 41x& = & 219 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{219}{41} \\\Leftrightarrow & x = \frac{219}{41} & & \\ & V = \left\{ \frac{219}{41} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+120& = & 65x-2340 \\\Leftrightarrow & 78x \color{red}{+120} \color{blue}{-120} \color{blue}{-65x} & = & \color{red}{65x} -2340 \color{blue}{-65x} \color{blue}{-120} \\\Leftrightarrow & 13x& = & -2460 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-2460}{13} \\\Leftrightarrow & x = \frac{-2460}{13} & & \\ & V = \left\{ \frac{-2460}{13} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 28x+336 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-28x} & = & \color{red}{28x} +336 \color{blue}{-28x} \color{blue}{+36} \\\Leftrightarrow & -7x& = & 372 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{372}{-7} \\\Leftrightarrow & x = \frac{-372}{7} & & \\ & V = \left\{ \frac{-372}{7} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x+98 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} +98 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 102 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{102}{-5} \\\Leftrightarrow & x = \frac{-102}{5} & & \\ & V = \left\{ \frac{-102}{5} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{7}{2}x+2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 10 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x+\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-10& = & 49x+28 \\\Leftrightarrow & 2x \color{red}{-10} \color{blue}{+10} \color{blue}{-49x} & = & \color{red}{49x} +28 \color{blue}{-49x} \color{blue}{+10} \\\Leftrightarrow & -47x& = & 38 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{38}{-47} \\\Leftrightarrow & x = \frac{-38}{47} & & \\ & V = \left\{ \frac{-38}{47} \right\} & \\\end{align}\)
9. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{8}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }- \frac{ 35 }{ \color{blue}{56} })& = & (\frac{196}{ \color{blue}{56} }x-\frac{168}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x-35& = & 196x-168 \\\Leftrightarrow & 8x \color{red}{-35} \color{blue}{+35} \color{blue}{-196x} & = & \color{red}{196x} -168 \color{blue}{-196x} \color{blue}{+35} \\\Leftrightarrow & -188x& = & -133 \\\Leftrightarrow & \frac{-188x}{ \color{red}{-188} }& = & \frac{-133}{-188} \\\Leftrightarrow & x = \frac{133}{188} & & \\ & V = \left\{ \frac{133}{188} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & -104x-312 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+104x} & = & \color{red}{-104x} -312 \color{blue}{+104x} \color{blue}{+24} \\\Leftrightarrow & 143x& = & -288 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-288}{143} \\\Leftrightarrow & x = \frac{-288}{143} & & \\ & V = \left\{ \frac{-288}{143} \right\} & \\\end{align}\)
11. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{441}{ \color{blue}{126} }x-\frac{630}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 441x-630 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-441x} & = & \color{red}{441x} -630 \color{blue}{-441x} \color{blue}{-56} \\\Leftrightarrow & -423x& = & -686 \\\Leftrightarrow & \frac{-423x}{ \color{red}{-423} }& = & \frac{-686}{-423} \\\Leftrightarrow & x = \frac{686}{423} & & \\ & V = \left\{ \frac{686}{423} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-30& = & 175x-420 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-175x} & = & \color{red}{175x} -420 \color{blue}{-175x} \color{blue}{+30} \\\Leftrightarrow & -161x& = & -390 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-390}{-161} \\\Leftrightarrow & x = \frac{390}{161} & & \\ & V = \left\{ \frac{390}{161} \right\} & \\\end{align}\)