Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 154x-396 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-154x} & = & \color{red}{154x} -396 \color{blue}{-154x} \color{blue}{+30} \\\Leftrightarrow & -121x& = & -366 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-366}{-121} \\\Leftrightarrow & x = \frac{366}{121} & & \\ & V = \left\{ \frac{366}{121} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{7}{6}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{35}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 35x-30 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-35x} & = & \color{red}{35x} -30 \color{blue}{-35x} \color{blue}{+8} \\\Leftrightarrow & -29x& = & -22 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-22}{-29} \\\Leftrightarrow & x = \frac{22}{29} & & \\ & V = \left\{ \frac{22}{29} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{-280}{ \color{blue}{105} }x+\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & -280x+105 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{+280x} & = & \color{red}{-280x} +105 \color{blue}{+280x} \color{blue}{-28} \\\Leftrightarrow & 295x& = & 77 \\\Leftrightarrow & \frac{295x}{ \color{red}{295} }& = & \frac{77}{295} \\\Leftrightarrow & x = \frac{77}{295} & & \\ & V = \left\{ \frac{77}{295} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{6}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 252x-1260 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-252x} & = & \color{red}{252x} -1260 \color{blue}{-252x} \color{blue}{-90} \\\Leftrightarrow & -217x& = & -1350 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-1350}{-217} \\\Leftrightarrow & x = \frac{1350}{217} & & \\ & V = \left\{ \frac{1350}{217} \right\} & \\\end{align}\)
5. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{60}{ \color{blue}{45} }x+\frac{45}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-20& = & 60x+45 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{-60x} & = & \color{red}{60x} +45 \color{blue}{-60x} \color{blue}{+20} \\\Leftrightarrow & -51x& = & 65 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{65}{-51} \\\Leftrightarrow & x = \frac{-65}{51} & & \\ & V = \left\{ \frac{-65}{51} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{7}{4}x+5 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{63}{ \color{blue}{36} }x+\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & 63x+180 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-63x} & = & \color{red}{63x} +180 \color{blue}{-63x} \color{blue}{-8} \\\Leftrightarrow & -51x& = & 172 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{172}{-51} \\\Leftrightarrow & x = \frac{-172}{51} & & \\ & V = \left\{ \frac{-172}{51} \right\} & \\\end{align}\)
7. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 21 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x+\frac{672}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-21& = & 56x+672 \\\Leftrightarrow & 16x \color{red}{-21} \color{blue}{+21} \color{blue}{-56x} & = & \color{red}{56x} +672 \color{blue}{-56x} \color{blue}{+21} \\\Leftrightarrow & -40x& = & 693 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{693}{-40} \\\Leftrightarrow & x = \frac{-693}{40} & & \\ & V = \left\{ \frac{-693}{40} \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x+75& = & 288x+960 \\\Leftrightarrow & 80x \color{red}{+75} \color{blue}{-75} \color{blue}{-288x} & = & \color{red}{288x} +960 \color{blue}{-288x} \color{blue}{-75} \\\Leftrightarrow & -208x& = & 885 \\\Leftrightarrow & \frac{-208x}{ \color{red}{-208} }& = & \frac{885}{-208} \\\Leftrightarrow & x = \frac{-885}{208} & & \\ & V = \left\{ \frac{-885}{208} \right\} & \\\end{align}\)
9. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 12x-72 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} -72 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & -4x& = & -63 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-63}{-4} \\\Leftrightarrow & x = \frac{63}{4} & & \\ & V = \left\{ \frac{63}{4} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 196x-84 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-196x} & = & \color{red}{196x} -84 \color{blue}{-196x} \color{blue}{-48} \\\Leftrightarrow & -175x& = & -132 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-132}{-175} \\\Leftrightarrow & x = \frac{132}{175} & & \\ & V = \left\{ \frac{132}{175} \right\} & \\\end{align}\)
11. \(\text{420 is het kleinste gemene veelvoud van 7, 12 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 175 }{ \color{blue}{420} })& = & (\frac{-588}{ \color{blue}{420} }x-\frac{420}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+175& = & -588x-420 \\\Leftrightarrow & 60x \color{red}{+175} \color{blue}{-175} \color{blue}{+588x} & = & \color{red}{-588x} -420 \color{blue}{+588x} \color{blue}{-175} \\\Leftrightarrow & 648x& = & -595 \\\Leftrightarrow & \frac{648x}{ \color{red}{648} }& = & \frac{-595}{648} \\\Leftrightarrow & x = \frac{-595}{648} & & \\ & V = \left\{ \frac{-595}{648} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{3}{4}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+36& = & 63x-168 \\\Leftrightarrow & 28x \color{red}{+36} \color{blue}{-36} \color{blue}{-63x} & = & \color{red}{63x} -168 \color{blue}{-63x} \color{blue}{-36} \\\Leftrightarrow & -35x& = & -204 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-204}{-35} \\\Leftrightarrow & x = \frac{204}{35} & & \\ & V = \left\{ \frac{204}{35} \right\} & \\\end{align}\)