Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -56x-588 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+56x} & = & \color{red}{-56x} -588 \color{blue}{+56x} \color{blue}{+48} \\\Leftrightarrow & 77x& = & -540 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-540}{77} \\\Leftrightarrow & x = \frac{-540}{77} & & \\ & V = \left\{ \frac{-540}{77} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x+280 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} +280 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & 260 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{260}{-21} \\\Leftrightarrow & x = \frac{-260}{21} & & \\ & V = \left\{ \frac{-260}{21} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -70x-294 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+70x} & = & \color{red}{-70x} -294 \color{blue}{+70x} \color{blue}{-12} \\\Leftrightarrow & 91x& = & -306 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-306}{91} \\\Leftrightarrow & x = \frac{-306}{91} & & \\ & V = \left\{ \frac{-306}{91} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{3}{5}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{36}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 36x-180 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-36x} & = & \color{red}{36x} -180 \color{blue}{-36x} \color{blue}{-8} \\\Leftrightarrow & -21x& = & -188 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-188}{-21} \\\Leftrightarrow & x = \frac{188}{21} & & \\ & V = \left\{ \frac{188}{21} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{2}{5}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{32}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+15& = & 32x+560 \\\Leftrightarrow & 40x \color{red}{+15} \color{blue}{-15} \color{blue}{-32x} & = & \color{red}{32x} +560 \color{blue}{-32x} \color{blue}{-15} \\\Leftrightarrow & 8x& = & 545 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{545}{8} \\\Leftrightarrow & x = \frac{545}{8} & & \\ & V = \left\{ \frac{545}{8} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{48}{ \color{blue}{80} }x+\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & 48x+160 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-48x} & = & \color{red}{48x} +160 \color{blue}{-48x} \color{blue}{-15} \\\Leftrightarrow & -28x& = & 145 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{145}{-28} \\\Leftrightarrow & x = \frac{-145}{28} & & \\ & V = \left\{ \frac{-145}{28} \right\} & \\\end{align}\)
7. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x-75& = & -96x-480 \\\Leftrightarrow & 80x \color{red}{-75} \color{blue}{+75} \color{blue}{+96x} & = & \color{red}{-96x} -480 \color{blue}{+96x} \color{blue}{+75} \\\Leftrightarrow & 176x& = & -405 \\\Leftrightarrow & \frac{176x}{ \color{red}{176} }& = & \frac{-405}{176} \\\Leftrightarrow & x = \frac{-405}{176} & & \\ & V = \left\{ \frac{-405}{176} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & 208x-780 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{-208x} & = & \color{red}{208x} -780 \color{blue}{-208x} \color{blue}{+60} \\\Leftrightarrow & -169x& = & -720 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-720}{-169} \\\Leftrightarrow & x = \frac{720}{169} & & \\ & V = \left\{ \frac{720}{169} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{42}{ \color{blue}{18} }x+\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 42x+90 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-42x} & = & \color{red}{42x} +90 \color{blue}{-42x} \color{blue}{-4} \\\Leftrightarrow & -33x& = & 86 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{86}{-33} \\\Leftrightarrow & x = \frac{-86}{33} & & \\ & V = \left\{ \frac{-86}{33} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x-720 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} -720 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & -760 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-760}{-3} \\\Leftrightarrow & x = \frac{760}{3} & & \\ & V = \left\{ \frac{760}{3} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-150& = & 55x+330 \\\Leftrightarrow & 66x \color{red}{-150} \color{blue}{+150} \color{blue}{-55x} & = & \color{red}{55x} +330 \color{blue}{-55x} \color{blue}{+150} \\\Leftrightarrow & 11x& = & 480 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{480}{11} \\\Leftrightarrow & x = \frac{480}{11} & & \\ & V = \left\{ \frac{480}{11} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{117}{ \color{blue}{78} }x+\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+18& = & 117x+468 \\\Leftrightarrow & 26x \color{red}{+18} \color{blue}{-18} \color{blue}{-117x} & = & \color{red}{117x} +468 \color{blue}{-117x} \color{blue}{-18} \\\Leftrightarrow & -91x& = & 450 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{450}{-91} \\\Leftrightarrow & x = \frac{-450}{91} & & \\ & V = \left\{ \frac{-450}{91} \right\} & \\\end{align}\)