Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 168x-700 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-168x} & = & \color{red}{168x} -700 \color{blue}{-168x} \color{blue}{-40} \\\Leftrightarrow & -133x& = & -740 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-740}{-133} \\\Leftrightarrow & x = \frac{740}{133} & & \\ & V = \left\{ \frac{740}{133} \right\} & \\\end{align}\)
2. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 56 }{ \color{blue}{154} })& = & (\frac{231}{ \color{blue}{154} }x-\frac{924}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+56& = & 231x-924 \\\Leftrightarrow & 22x \color{red}{+56} \color{blue}{-56} \color{blue}{-231x} & = & \color{red}{231x} -924 \color{blue}{-231x} \color{blue}{-56} \\\Leftrightarrow & -209x& = & -980 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-980}{-209} \\\Leftrightarrow & x = \frac{980}{209} & & \\ & V = \left\{ \frac{980}{209} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{11}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-18& = & 154x+198 \\\Leftrightarrow & 33x \color{red}{-18} \color{blue}{+18} \color{blue}{-154x} & = & \color{red}{154x} +198 \color{blue}{-154x} \color{blue}{+18} \\\Leftrightarrow & -121x& = & 216 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{216}{-121} \\\Leftrightarrow & x = \frac{-216}{121} & & \\ & V = \left\{ \frac{-216}{121} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & -462x-990 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{+462x} & = & \color{red}{-462x} -990 \color{blue}{+462x} \color{blue}{-120} \\\Leftrightarrow & 517x& = & -1110 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-1110}{517} \\\Leftrightarrow & x = \frac{-1110}{517} & & \\ & V = \left\{ \frac{-1110}{517} \right\} & \\\end{align}\)
5. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{8}& = & \frac{3}{2}x-8 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 15 }{ \color{blue}{40} })& = & (\frac{60}{ \color{blue}{40} }x-\frac{320}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+15& = & 60x-320 \\\Leftrightarrow & 8x \color{red}{+15} \color{blue}{-15} \color{blue}{-60x} & = & \color{red}{60x} -320 \color{blue}{-60x} \color{blue}{-15} \\\Leftrightarrow & -52x& = & -335 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{-335}{-52} \\\Leftrightarrow & x = \frac{335}{52} & & \\ & V = \left\{ \frac{335}{52} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{3}{2}x+3 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{42}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 21x+42 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-21x} & = & \color{red}{21x} +42 \color{blue}{-21x} \color{blue}{-6} \\\Leftrightarrow & -19x& = & 36 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{36}{-19} \\\Leftrightarrow & x = \frac{-36}{19} & & \\ & V = \left\{ \frac{-36}{19} \right\} & \\\end{align}\)
7. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{280}{ \color{blue}{112} }x-\frac{672}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 280x-672 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-280x} & = & \color{red}{280x} -672 \color{blue}{-280x} \color{blue}{-35} \\\Leftrightarrow & -264x& = & -707 \\\Leftrightarrow & \frac{-264x}{ \color{red}{-264} }& = & \frac{-707}{-264} \\\Leftrightarrow & x = \frac{707}{264} & & \\ & V = \left\{ \frac{707}{264} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{4}{5}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{56}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & 56x-140 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{-56x} & = & \color{red}{56x} -140 \color{blue}{-56x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & -120 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-120}{-21} \\\Leftrightarrow & x = \frac{40}{7} & & \\ & V = \left\{ \frac{40}{7} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 65x+1170 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} +1170 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & 13x& = & 1110 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{1110}{13} \\\Leftrightarrow & x = \frac{1110}{13} & & \\ & V = \left\{ \frac{1110}{13} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 65x-650 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} -650 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -39x& = & -690 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-690}{-39} \\\Leftrightarrow & x = \frac{230}{13} & & \\ & V = \left\{ \frac{230}{13} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & -117x+156 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{+117x} & = & \color{red}{-117x} +156 \color{blue}{+117x} \color{blue}{+24} \\\Leftrightarrow & 169x& = & 180 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{180}{169} \\\Leftrightarrow & x = \frac{180}{169} & & \\ & V = \left\{ \frac{180}{169} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x+\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & -52x+260 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{+52x} & = & \color{red}{-52x} +260 \color{blue}{+52x} \color{blue}{+20} \\\Leftrightarrow & 117x& = & 280 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{280}{117} \\\Leftrightarrow & x = \frac{280}{117} & & \\ & V = \left\{ \frac{280}{117} \right\} & \\\end{align}\)