Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x+\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-24& = & -88x+132 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+88x} & = & \color{red}{-88x} +132 \color{blue}{+88x} \color{blue}{+24} \\\Leftrightarrow & 121x& = & 156 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{156}{121} \\\Leftrightarrow & x = \frac{156}{121} & & \\ & V = \left\{ \frac{156}{121} \right\} & \\\end{align}\)
2. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 21 }{ \color{blue}{56} })& = & (\frac{28}{ \color{blue}{56} }x-\frac{56}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+21& = & 28x-56 \\\Leftrightarrow & 8x \color{red}{+21} \color{blue}{-21} \color{blue}{-28x} & = & \color{red}{28x} -56 \color{blue}{-28x} \color{blue}{-21} \\\Leftrightarrow & -20x& = & -77 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{-77}{-20} \\\Leftrightarrow & x = \frac{77}{20} & & \\ & V = \left\{ \frac{77}{20} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 6x-30 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} -30 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & 9x& = & -26 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{-26}{9} \\\Leftrightarrow & x = \frac{-26}{9} & & \\ & V = \left\{ \frac{-26}{9} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{4}x-2 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x-\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 33x-264 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-33x} & = & \color{red}{33x} -264 \color{blue}{-33x} \color{blue}{+60} \\\Leftrightarrow & 11x& = & -204 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-204}{11} \\\Leftrightarrow & x = \frac{-204}{11} & & \\ & V = \left\{ \frac{-204}{11} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -84x-840 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} -840 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & -900 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-900}{119} \\\Leftrightarrow & x = \frac{-900}{119} & & \\ & V = \left\{ \frac{-900}{119} \right\} & \\\end{align}\)
6. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }- \frac{ 112 }{ \color{blue}{252} })& = & (\frac{-441}{ \color{blue}{252} }x+\frac{1008}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x-112& = & -441x+1008 \\\Leftrightarrow & 36x \color{red}{-112} \color{blue}{+112} \color{blue}{+441x} & = & \color{red}{-441x} +1008 \color{blue}{+441x} \color{blue}{+112} \\\Leftrightarrow & 477x& = & 1120 \\\Leftrightarrow & \frac{477x}{ \color{red}{477} }& = & \frac{1120}{477} \\\Leftrightarrow & x = \frac{1120}{477} & & \\ & V = \left\{ \frac{1120}{477} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x-\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 49x-98 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-49x} & = & \color{red}{49x} -98 \color{blue}{-49x} \color{blue}{+4} \\\Leftrightarrow & -47x& = & -94 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{-94}{-47} \\\Leftrightarrow & x = 2 & & \\ & V = \left\{ 2 \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x-\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 35x-56 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-35x} & = & \color{red}{35x} -56 \color{blue}{-35x} \color{blue}{-6} \\\Leftrightarrow & -33x& = & -62 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-62}{-33} \\\Leftrightarrow & x = \frac{62}{33} & & \\ & V = \left\{ \frac{62}{33} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & 16x+96 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{-16x} & = & \color{red}{16x} +96 \color{blue}{-16x} \color{blue}{+9} \\\Leftrightarrow & 8x& = & 105 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{105}{8} \\\Leftrightarrow & x = \frac{105}{8} & & \\ & V = \left\{ \frac{105}{8} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & -40x+60 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+40x} & = & \color{red}{-40x} +60 \color{blue}{+40x} \color{blue}{+8} \\\Leftrightarrow & 55x& = & 68 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{68}{55} \\\Leftrightarrow & x = \frac{68}{55} & & \\ & V = \left\{ \frac{68}{55} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{308}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & 308x-1056 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{-308x} & = & \color{red}{308x} -1056 \color{blue}{-308x} \color{blue}{-48} \\\Leftrightarrow & -275x& = & -1104 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{-1104}{-275} \\\Leftrightarrow & x = \frac{1104}{275} & & \\ & V = \left\{ \frac{1104}{275} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-126 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -126 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -150 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-150}{-7} \\\Leftrightarrow & x = \frac{150}{7} & & \\ & V = \left\{ \frac{150}{7} \right\} & \\\end{align}\)