Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{-154}{ \color{blue}{110} }x+\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+20& = & -154x+550 \\\Leftrightarrow & 55x \color{red}{+20} \color{blue}{-20} \color{blue}{+154x} & = & \color{red}{-154x} +550 \color{blue}{+154x} \color{blue}{-20} \\\Leftrightarrow & 209x& = & 530 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{530}{209} \\\Leftrightarrow & x = \frac{530}{209} & & \\ & V = \left\{ \frac{530}{209} \right\} & \\\end{align}\)
2. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & -462x+990 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{+462x} & = & \color{red}{-462x} +990 \color{blue}{+462x} \color{blue}{-120} \\\Leftrightarrow & 517x& = & 870 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{870}{517} \\\Leftrightarrow & x = \frac{870}{517} & & \\ & V = \left\{ \frac{870}{517} \right\} & \\\end{align}\)
3. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 42 }{ \color{blue}{273} })& = & (\frac{364}{ \color{blue}{273} }x+\frac{546}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-42& = & 364x+546 \\\Leftrightarrow & 39x \color{red}{-42} \color{blue}{+42} \color{blue}{-364x} & = & \color{red}{364x} +546 \color{blue}{-364x} \color{blue}{+42} \\\Leftrightarrow & -325x& = & 588 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{588}{-325} \\\Leftrightarrow & x = \frac{-588}{325} & & \\ & V = \left\{ \frac{-588}{325} \right\} & \\\end{align}\)
4. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{7}{ \color{blue}{21} }x+\frac{126}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & 7x+126 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{-7x} & = & \color{red}{7x} +126 \color{blue}{-7x} \color{blue}{+12} \\\Leftrightarrow & -4x& = & 138 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{138}{-4} \\\Leftrightarrow & x = \frac{-69}{2} & & \\ & V = \left\{ \frac{-69}{2} \right\} & \\\end{align}\)
5. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 105 }{ \color{blue}{231} })& = & (\frac{-154}{ \color{blue}{231} }x-\frac{924}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-105& = & -154x-924 \\\Leftrightarrow & 33x \color{red}{-105} \color{blue}{+105} \color{blue}{+154x} & = & \color{red}{-154x} -924 \color{blue}{+154x} \color{blue}{+105} \\\Leftrightarrow & 187x& = & -819 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-819}{187} \\\Leftrightarrow & x = \frac{-819}{187} & & \\ & V = \left\{ \frac{-819}{187} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{6}{5}x-6 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x-\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & 96x-480 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{-96x} & = & \color{red}{96x} -480 \color{blue}{-96x} \color{blue}{+25} \\\Leftrightarrow & -76x& = & -455 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{-455}{-76} \\\Leftrightarrow & x = \frac{455}{76} & & \\ & V = \left\{ \frac{455}{76} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & -294x+1680 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{+294x} & = & \color{red}{-294x} +1680 \color{blue}{+294x} \color{blue}{+120} \\\Leftrightarrow & 329x& = & 1800 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{1800}{329} \\\Leftrightarrow & x = \frac{1800}{329} & & \\ & V = \left\{ \frac{1800}{329} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{2}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{28}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & 28x+350 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-28x} & = & \color{red}{28x} +350 \color{blue}{-28x} \color{blue}{-40} \\\Leftrightarrow & 7x& = & 310 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{310}{7} \\\Leftrightarrow & x = \frac{310}{7} & & \\ & V = \left\{ \frac{310}{7} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{30}{ \color{blue}{60} }x+\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 30x+300 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-30x} & = & \color{red}{30x} +300 \color{blue}{-30x} \color{blue}{+25} \\\Leftrightarrow & -18x& = & 325 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{325}{-18} \\\Leftrightarrow & x = \frac{-325}{18} & & \\ & V = \left\{ \frac{-325}{18} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x+360 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} +360 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & 400 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{400}{-27} \\\Leftrightarrow & x = \frac{-400}{27} & & \\ & V = \left\{ \frac{-400}{27} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -20x+60 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+20x} & = & \color{red}{-20x} +60 \color{blue}{+20x} \color{blue}{+8} \\\Leftrightarrow & 35x& = & 68 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{68}{35} \\\Leftrightarrow & x = \frac{68}{35} & & \\ & V = \left\{ \frac{68}{35} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 15x+150 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-15x} & = & \color{red}{15x} +150 \color{blue}{-15x} \color{blue}{-25} \\\Leftrightarrow & -9x& = & 125 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{125}{-9} \\\Leftrightarrow & x = \frac{-125}{9} & & \\ & V = \left\{ \frac{-125}{9} \right\} & \\\end{align}\)