Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}-\frac{2}{7}=\frac{1}{6}x+7\)
2. \(\frac{x}{5}+\frac{3}{13}=\frac{1}{2}x-2\)
3. \(\frac{x}{4}+\frac{4}{15}=\frac{-2}{3}x-4\)
4. \(\frac{x}{4}-\frac{5}{16}=\frac{1}{3}x+7\)
5. \(\frac{x}{3}+\frac{5}{16}=\frac{5}{2}x+8\)
6. \(\frac{x}{5}-\frac{4}{11}=\frac{3}{2}x+8\)
7. \(\frac{x}{4}+\frac{3}{14}=\frac{1}{3}x-3\)
8. \(\frac{x}{6}-\frac{2}{11}=\frac{4}{5}x-5\)
9. \(\frac{x}{2}+\frac{5}{16}=\frac{-2}{3}x+3\)
10. \(\frac{x}{7}+\frac{5}{14}=\frac{1}{2}x-2\)
11. \(\frac{x}{4}-\frac{4}{11}=\frac{-2}{3}x-7\)
12. \(\frac{x}{7}-\frac{3}{11}=\frac{1}{2}x-7\)

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{6}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-12& = & 7x+294 \\\Leftrightarrow & 6x \color{red}{-12} \color{blue}{+12} \color{blue}{-7x} & = & \color{red}{7x} +294 \color{blue}{-7x} \color{blue}{+12} \\\Leftrightarrow & -x& = & 306 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{306}{-1} \\\Leftrightarrow & x = -306 & & \\ & V = \left\{ -306 \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x-260 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} -260 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & -290 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-290}{-39} \\\Leftrightarrow & x = \frac{290}{39} & & \\ & V = \left\{ \frac{290}{39} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -40x-240 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+40x} & = & \color{red}{-40x} -240 \color{blue}{+40x} \color{blue}{-16} \\\Leftrightarrow & 55x& = & -256 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{-256}{55} \\\Leftrightarrow & x = \frac{-256}{55} & & \\ & V = \left\{ \frac{-256}{55} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-15& = & 16x+336 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} +336 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & 351 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{351}{-4} \\\Leftrightarrow & x = \frac{-351}{4} & & \\ & V = \left\{ \frac{-351}{4} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{5}{2}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{120}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & 120x+384 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-120x} & = & \color{red}{120x} +384 \color{blue}{-120x} \color{blue}{-15} \\\Leftrightarrow & -104x& = & 369 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{369}{-104} \\\Leftrightarrow & x = \frac{-369}{104} & & \\ & V = \left\{ \frac{-369}{104} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{165}{ \color{blue}{110} }x+\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 165x+880 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-165x} & = & \color{red}{165x} +880 \color{blue}{-165x} \color{blue}{+40} \\\Leftrightarrow & -143x& = & 920 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{920}{-143} \\\Leftrightarrow & x = \frac{-920}{143} & & \\ & V = \left\{ \frac{-920}{143} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 18 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+18& = & 28x-252 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{-28x} & = & \color{red}{28x} -252 \color{blue}{-28x} \color{blue}{-18} \\\Leftrightarrow & -7x& = & -270 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-270}{-7} \\\Leftrightarrow & x = \frac{270}{7} & & \\ & V = \left\{ \frac{270}{7} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{264}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & 264x-1650 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-264x} & = & \color{red}{264x} -1650 \color{blue}{-264x} \color{blue}{+60} \\\Leftrightarrow & -209x& = & -1590 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-1590}{-209} \\\Leftrightarrow & x = \frac{1590}{209} & & \\ & V = \left\{ \frac{1590}{209} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & -32x+144 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{+32x} & = & \color{red}{-32x} +144 \color{blue}{+32x} \color{blue}{-15} \\\Leftrightarrow & 56x& = & 129 \\\Leftrightarrow & \frac{56x}{ \color{red}{56} }& = & \frac{129}{56} \\\Leftrightarrow & x = \frac{129}{56} & & \\ & V = \left\{ \frac{129}{56} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 5 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+5& = & 7x-28 \\\Leftrightarrow & 2x \color{red}{+5} \color{blue}{-5} \color{blue}{-7x} & = & \color{red}{7x} -28 \color{blue}{-7x} \color{blue}{-5} \\\Leftrightarrow & -5x& = & -33 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-33}{-5} \\\Leftrightarrow & x = \frac{33}{5} & & \\ & V = \left\{ \frac{33}{5} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & -88x-924 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{+88x} & = & \color{red}{-88x} -924 \color{blue}{+88x} \color{blue}{+48} \\\Leftrightarrow & 121x& = & -876 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-876}{121} \\\Leftrightarrow & x = \frac{-876}{121} & & \\ & V = \left\{ \frac{-876}{121} \right\} & \\\end{align}\)
12. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 42 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-42& = & 77x-1078 \\\Leftrightarrow & 22x \color{red}{-42} \color{blue}{+42} \color{blue}{-77x} & = & \color{red}{77x} -1078 \color{blue}{-77x} \color{blue}{+42} \\\Leftrightarrow & -55x& = & -1036 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-1036}{-55} \\\Leftrightarrow & x = \frac{1036}{55} & & \\ & V = \left\{ \frac{1036}{55} \right\} & \\\end{align}\)