Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+60& = & 35x+280 \\\Leftrightarrow & 28x \color{red}{+60} \color{blue}{-60} \color{blue}{-35x} & = & \color{red}{35x} +280 \color{blue}{-35x} \color{blue}{-60} \\\Leftrightarrow & -7x& = & 220 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{220}{-7} \\\Leftrightarrow & x = \frac{-220}{7} & & \\ & V = \left\{ \frac{-220}{7} \right\} & \\\end{align}\)
2. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-42}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+30& = & -42x-630 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{+42x} & = & \color{red}{-42x} -630 \color{blue}{+42x} \color{blue}{-30} \\\Leftrightarrow & 77x& = & -660 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-660}{77} \\\Leftrightarrow & x = \frac{-60}{7} & & \\ & V = \left\{ \frac{-60}{7} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{2}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 12x-240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} -240 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & -7x& = & -232 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-232}{-7} \\\Leftrightarrow & x = \frac{232}{7} & & \\ & V = \left\{ \frac{232}{7} \right\} & \\\end{align}\)
4. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 77x+616 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-77x} & = & \color{red}{77x} +616 \color{blue}{-77x} \color{blue}{-28} \\\Leftrightarrow & -55x& = & 588 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{588}{-55} \\\Leftrightarrow & x = \frac{-588}{55} & & \\ & V = \left\{ \frac{-588}{55} \right\} & \\\end{align}\)
5. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{4}x+8 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x+\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+40& = & 65x+2080 \\\Leftrightarrow & 52x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} +2080 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -13x& = & 2040 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{2040}{-13} \\\Leftrightarrow & x = \frac{-2040}{13} & & \\ & V = \left\{ \frac{-2040}{13} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -70x+252 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+70x} & = & \color{red}{-70x} +252 \color{blue}{+70x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & 234 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{234}{91} \\\Leftrightarrow & x = \frac{18}{7} & & \\ & V = \left\{ \frac{18}{7} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 56 }{ \color{blue}{210} })& = & (\frac{525}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-56& = & 525x-1260 \\\Leftrightarrow & 30x \color{red}{-56} \color{blue}{+56} \color{blue}{-525x} & = & \color{red}{525x} -1260 \color{blue}{-525x} \color{blue}{+56} \\\Leftrightarrow & -495x& = & -1204 \\\Leftrightarrow & \frac{-495x}{ \color{red}{-495} }& = & \frac{-1204}{-495} \\\Leftrightarrow & x = \frac{1204}{495} & & \\ & V = \left\{ \frac{1204}{495} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{-5}{6}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & -275x+2640 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{+275x} & = & \color{red}{-275x} +2640 \color{blue}{+275x} \color{blue}{-150} \\\Leftrightarrow & 341x& = & 2490 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{2490}{341} \\\Leftrightarrow & x = \frac{2490}{341} & & \\ & V = \left\{ \frac{2490}{341} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & 168x+420 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{-168x} & = & \color{red}{168x} +420 \color{blue}{-168x} \color{blue}{+150} \\\Leftrightarrow & -133x& = & 570 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{570}{-133} \\\Leftrightarrow & x = \frac{-30}{7} & & \\ & V = \left\{ \frac{-30}{7} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & -42x-180 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+42x} & = & \color{red}{-42x} -180 \color{blue}{+42x} \color{blue}{-4} \\\Leftrightarrow & 47x& = & -184 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-184}{47} \\\Leftrightarrow & x = \frac{-184}{47} & & \\ & V = \left\{ \frac{-184}{47} \right\} & \\\end{align}\)
11. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{-7}{4}x-5 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-42}{ \color{blue}{24} }x-\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & -42x-120 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{+42x} & = & \color{red}{-42x} -120 \color{blue}{+42x} \color{blue}{+9} \\\Leftrightarrow & 50x& = & -111 \\\Leftrightarrow & \frac{50x}{ \color{red}{50} }& = & \frac{-111}{50} \\\Leftrightarrow & x = \frac{-111}{50} & & \\ & V = \left\{ \frac{-111}{50} \right\} & \\\end{align}\)
12. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 168 }{ \color{blue}{546} })& = & (\frac{-455}{ \color{blue}{546} }x+\frac{3276}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-168& = & -455x+3276 \\\Leftrightarrow & 78x \color{red}{-168} \color{blue}{+168} \color{blue}{+455x} & = & \color{red}{-455x} +3276 \color{blue}{+455x} \color{blue}{+168} \\\Leftrightarrow & 533x& = & 3444 \\\Leftrightarrow & \frac{533x}{ \color{red}{533} }& = & \frac{3444}{533} \\\Leftrightarrow & x = \frac{84}{13} & & \\ & V = \left\{ \frac{84}{13} \right\} & \\\end{align}\)