Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{7}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & 294x-1260 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{-294x} & = & \color{red}{294x} -1260 \color{blue}{-294x} \color{blue}{-75} \\\Leftrightarrow & -259x& = & -1335 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-1335}{-259} \\\Leftrightarrow & x = \frac{1335}{259} & & \\ & V = \left\{ \frac{1335}{259} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -156x-2730 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+156x} & = & \color{red}{-156x} -2730 \color{blue}{+156x} \color{blue}{-120} \\\Leftrightarrow & 221x& = & -2850 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-2850}{221} \\\Leftrightarrow & x = \frac{-2850}{221} & & \\ & V = \left\{ \frac{-2850}{221} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 105x-252 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-105x} & = & \color{red}{105x} -252 \color{blue}{-105x} \color{blue}{+12} \\\Leftrightarrow & -91x& = & -240 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-240}{-91} \\\Leftrightarrow & x = \frac{240}{91} & & \\ & V = \left\{ \frac{240}{91} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x-\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & -32x-288 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{+32x} & = & \color{red}{-32x} -288 \color{blue}{+32x} \color{blue}{-9} \\\Leftrightarrow & 44x& = & -297 \\\Leftrightarrow & \frac{44x}{ \color{red}{44} }& = & \frac{-297}{44} \\\Leftrightarrow & x = \frac{-27}{4} & & \\ & V = \left\{ \frac{-27}{4} \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{8}& = & \frac{2}{3}x-7 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{16}{ \color{blue}{24} }x-\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-9& = & 16x-168 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-16x} & = & \color{red}{16x} -168 \color{blue}{-16x} \color{blue}{+9} \\\Leftrightarrow & -10x& = & -159 \\\Leftrightarrow & \frac{-10x}{ \color{red}{-10} }& = & \frac{-159}{-10} \\\Leftrightarrow & x = \frac{159}{10} & & \\ & V = \left\{ \frac{159}{10} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-20}{ \color{blue}{12} }x+\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & -20x+48 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{+20x} & = & \color{red}{-20x} +48 \color{blue}{+20x} \color{blue}{+10} \\\Leftrightarrow & 23x& = & 58 \\\Leftrightarrow & \frac{23x}{ \color{red}{23} }& = & \frac{58}{23} \\\Leftrightarrow & x = \frac{58}{23} & & \\ & V = \left\{ \frac{58}{23} \right\} & \\\end{align}\)
7. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{30}{ \color{blue}{12} }x-\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-5& = & 30x-84 \\\Leftrightarrow & 4x \color{red}{-5} \color{blue}{+5} \color{blue}{-30x} & = & \color{red}{30x} -84 \color{blue}{-30x} \color{blue}{+5} \\\Leftrightarrow & -26x& = & -79 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-79}{-26} \\\Leftrightarrow & x = \frac{79}{26} & & \\ & V = \left\{ \frac{79}{26} \right\} & \\\end{align}\)
8. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{-49}{ \color{blue}{35} }x-\frac{175}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & -49x-175 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{+49x} & = & \color{red}{-49x} -175 \color{blue}{+49x} \color{blue}{-20} \\\Leftrightarrow & 54x& = & -195 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{-195}{54} \\\Leftrightarrow & x = \frac{-65}{18} & & \\ & V = \left\{ \frac{-65}{18} \right\} & \\\end{align}\)
9. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{320}{ \color{blue}{240} }x+\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & 320x+960 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{-320x} & = & \color{red}{320x} +960 \color{blue}{-320x} \color{blue}{-45} \\\Leftrightarrow & -272x& = & 915 \\\Leftrightarrow & \frac{-272x}{ \color{red}{-272} }& = & \frac{915}{-272} \\\Leftrightarrow & x = \frac{-915}{272} & & \\ & V = \left\{ \frac{-915}{272} \right\} & \\\end{align}\)
10. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-135}{ \color{blue}{180} }x+\frac{1260}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x+40& = & -135x+1260 \\\Leftrightarrow & 36x \color{red}{+40} \color{blue}{-40} \color{blue}{+135x} & = & \color{red}{-135x} +1260 \color{blue}{+135x} \color{blue}{-40} \\\Leftrightarrow & 171x& = & 1220 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{1220}{171} \\\Leftrightarrow & x = \frac{1220}{171} & & \\ & V = \left\{ \frac{1220}{171} \right\} & \\\end{align}\)
11. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{3}{4}x-1 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }+ \frac{ 56 }{ \color{blue}{364} })& = & (\frac{273}{ \color{blue}{364} }x-\frac{364}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x+56& = & 273x-364 \\\Leftrightarrow & 52x \color{red}{+56} \color{blue}{-56} \color{blue}{-273x} & = & \color{red}{273x} -364 \color{blue}{-273x} \color{blue}{-56} \\\Leftrightarrow & -221x& = & -420 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{-420}{-221} \\\Leftrightarrow & x = \frac{420}{221} & & \\ & V = \left\{ \frac{420}{221} \right\} & \\\end{align}\)
12. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{2}{5}x+2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 10 }{ \color{blue}{35} })& = & (\frac{14}{ \color{blue}{35} }x+\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+10& = & 14x+70 \\\Leftrightarrow & 5x \color{red}{+10} \color{blue}{-10} \color{blue}{-14x} & = & \color{red}{14x} +70 \color{blue}{-14x} \color{blue}{-10} \\\Leftrightarrow & -9x& = & 60 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{60}{-9} \\\Leftrightarrow & x = \frac{-20}{3} & & \\ & V = \left\{ \frac{-20}{3} \right\} & \\\end{align}\)