Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 105 }{ \color{blue}{231} })& = & (\frac{308}{ \color{blue}{231} }x+\frac{462}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+105& = & 308x+462 \\\Leftrightarrow & 33x \color{red}{+105} \color{blue}{-105} \color{blue}{-308x} & = & \color{red}{308x} +462 \color{blue}{-308x} \color{blue}{-105} \\\Leftrightarrow & -275x& = & 357 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{357}{-275} \\\Leftrightarrow & x = \frac{-357}{275} & & \\ & V = \left\{ \frac{-357}{275} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{12}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 12x+288 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} +288 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & 4x& = & 297 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{297}{4} \\\Leftrightarrow & x = \frac{297}{4} & & \\ & V = \left\{ \frac{297}{4} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-15& = & 96x+400 \\\Leftrightarrow & 20x \color{red}{-15} \color{blue}{+15} \color{blue}{-96x} & = & \color{red}{96x} +400 \color{blue}{-96x} \color{blue}{+15} \\\Leftrightarrow & -76x& = & 415 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{415}{-76} \\\Leftrightarrow & x = \frac{-415}{76} & & \\ & V = \left\{ \frac{-415}{76} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -196x-280 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+196x} & = & \color{red}{-196x} -280 \color{blue}{+196x} \color{blue}{-80} \\\Leftrightarrow & 231x& = & -360 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-360}{231} \\\Leftrightarrow & x = \frac{-120}{77} & & \\ & V = \left\{ \frac{-120}{77} \right\} & \\\end{align}\)
5. \(\text{280 is het kleinste gemene veelvoud van 7, 8 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{8}& = & \frac{2}{5}x-6 \\\Leftrightarrow & \color{blue}{280.} (\frac{40x}{ \color{blue}{280} }- \frac{ 175 }{ \color{blue}{280} })& = & (\frac{112}{ \color{blue}{280} }x-\frac{1680}{ \color{blue}{280} }) \color{blue}{.280} \\\Leftrightarrow & 40x-175& = & 112x-1680 \\\Leftrightarrow & 40x \color{red}{-175} \color{blue}{+175} \color{blue}{-112x} & = & \color{red}{112x} -1680 \color{blue}{-112x} \color{blue}{+175} \\\Leftrightarrow & -72x& = & -1505 \\\Leftrightarrow & \frac{-72x}{ \color{red}{-72} }& = & \frac{-1505}{-72} \\\Leftrightarrow & x = \frac{1505}{72} & & \\ & V = \left\{ \frac{1505}{72} \right\} & \\\end{align}\)
6. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{-36}{ \color{blue}{45} }x+\frac{225}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-20& = & -36x+225 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{+36x} & = & \color{red}{-36x} +225 \color{blue}{+36x} \color{blue}{+20} \\\Leftrightarrow & 51x& = & 245 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{245}{51} \\\Leftrightarrow & x = \frac{245}{51} & & \\ & V = \left\{ \frac{245}{51} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+30& = & 21x-210 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{-30} \\\Leftrightarrow & -7x& = & -240 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-240}{-7} \\\Leftrightarrow & x = \frac{240}{7} & & \\ & V = \left\{ \frac{240}{7} \right\} & \\\end{align}\)
8. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 168 }{ \color{blue}{546} })& = & (\frac{91}{ \color{blue}{546} }x+\frac{1638}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+168& = & 91x+1638 \\\Leftrightarrow & 78x \color{red}{+168} \color{blue}{-168} \color{blue}{-91x} & = & \color{red}{91x} +1638 \color{blue}{-91x} \color{blue}{-168} \\\Leftrightarrow & -13x& = & 1470 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1470}{-13} \\\Leftrightarrow & x = \frac{-1470}{13} & & \\ & V = \left\{ \frac{-1470}{13} \right\} & \\\end{align}\)
9. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x+\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 40x+320 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-40x} & = & \color{red}{40x} +320 \color{blue}{-40x} \color{blue}{-25} \\\Leftrightarrow & -24x& = & 295 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{295}{-24} \\\Leftrightarrow & x = \frac{-295}{24} & & \\ & V = \left\{ \frac{-295}{24} \right\} & \\\end{align}\)
10. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x+\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & 35x+420 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{-35x} & = & \color{red}{35x} +420 \color{blue}{-35x} \color{blue}{-28} \\\Leftrightarrow & -20x& = & 392 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{392}{-20} \\\Leftrightarrow & x = \frac{-98}{5} & & \\ & V = \left\{ \frac{-98}{5} \right\} & \\\end{align}\)
11. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-252}{ \color{blue}{180} }x+\frac{1260}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & -252x+1260 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{+252x} & = & \color{red}{-252x} +1260 \color{blue}{+252x} \color{blue}{-40} \\\Leftrightarrow & 297x& = & 1220 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{1220}{297} \\\Leftrightarrow & x = \frac{1220}{297} & & \\ & V = \left\{ \frac{1220}{297} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{5}{6}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 63 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-63& = & 175x+1470 \\\Leftrightarrow & 30x \color{red}{-63} \color{blue}{+63} \color{blue}{-175x} & = & \color{red}{175x} +1470 \color{blue}{-175x} \color{blue}{+63} \\\Leftrightarrow & -145x& = & 1533 \\\Leftrightarrow & \frac{-145x}{ \color{red}{-145} }& = & \frac{1533}{-145} \\\Leftrightarrow & x = \frac{-1533}{145} & & \\ & V = \left\{ \frac{-1533}{145} \right\} & \\\end{align}\)