Vgln. eerste graad (reeks 4)

Hoofdmenu Eentje per keer  🖨

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

1. \(\frac{x}{2}-\frac{4}{15}=\frac{-7}{5}x+8\)
2. \(\frac{x}{7}+\frac{3}{7}=\frac{5}{2}x+2\)
3. \(\frac{x}{5}+\frac{3}{16}=\frac{1}{6}x+5\)
4. \(\frac{x}{6}-\frac{2}{15}=\frac{1}{5}x+3\)
5. \(\frac{x}{2}-\frac{2}{7}=\frac{-7}{5}x-7\)
6. \(\frac{x}{7}+\frac{4}{7}=\frac{5}{3}x+5\)
7. \(\frac{x}{5}-\frac{5}{16}=\frac{3}{2}x-2\)
8. \(\frac{x}{5}-\frac{3}{11}=\frac{5}{6}x-6\)
9. \(\frac{x}{3}+\frac{5}{8}=\frac{-3}{4}x-2\)
10. \(\frac{x}{2}+\frac{3}{7}=\frac{-5}{3}x-3\)
11. \(\frac{x}{2}+\frac{2}{9}=\frac{1}{3}x-7\)
12. \(\frac{x}{3}-\frac{3}{14}=\frac{1}{5}x+2\)

---

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -42x+240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} +240 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 57x& = & 248 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{248}{57} \\\Leftrightarrow & x = \frac{248}{57} & & \\ & V = \left\{ \frac{248}{57} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{5}{2}x+2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x+\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 35x+28 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-35x} & = & \color{red}{35x} +28 \color{blue}{-35x} \color{blue}{-6} \\\Leftrightarrow & -33x& = & 22 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{22}{-33} \\\Leftrightarrow & x = \frac{-2}{3} & & \\ & V = \left\{ \frac{-2}{3} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x+\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & 40x+1200 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{-40x} & = & \color{red}{40x} +1200 \color{blue}{-40x} \color{blue}{-45} \\\Leftrightarrow & 8x& = & 1155 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{1155}{8} \\\Leftrightarrow & x = \frac{1155}{8} & & \\ & V = \left\{ \frac{1155}{8} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 6x+90 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} +90 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & -x& = & 94 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{94}{-1} \\\Leftrightarrow & x = -94 & & \\ & V = \left\{ -94 \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -98x-490 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+98x} & = & \color{red}{-98x} -490 \color{blue}{+98x} \color{blue}{+20} \\\Leftrightarrow & 133x& = & -470 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-470}{133} \\\Leftrightarrow & x = \frac{-470}{133} & & \\ & V = \left\{ \frac{-470}{133} \right\} & \\\end{align}\)
6. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{5}{3}x+5 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 12 }{ \color{blue}{21} })& = & (\frac{35}{ \color{blue}{21} }x+\frac{105}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+12& = & 35x+105 \\\Leftrightarrow & 3x \color{red}{+12} \color{blue}{-12} \color{blue}{-35x} & = & \color{red}{35x} +105 \color{blue}{-35x} \color{blue}{-12} \\\Leftrightarrow & -32x& = & 93 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{93}{-32} \\\Leftrightarrow & x = \frac{-93}{32} & & \\ & V = \left\{ \frac{-93}{32} \right\} & \\\end{align}\)
7. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{120}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 120x-160 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-120x} & = & \color{red}{120x} -160 \color{blue}{-120x} \color{blue}{+25} \\\Leftrightarrow & -104x& = & -135 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{-135}{-104} \\\Leftrightarrow & x = \frac{135}{104} & & \\ & V = \left\{ \frac{135}{104} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{11}& = & \frac{5}{6}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-90& = & 275x-1980 \\\Leftrightarrow & 66x \color{red}{-90} \color{blue}{+90} \color{blue}{-275x} & = & \color{red}{275x} -1980 \color{blue}{-275x} \color{blue}{+90} \\\Leftrightarrow & -209x& = & -1890 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-1890}{-209} \\\Leftrightarrow & x = \frac{1890}{209} & & \\ & V = \left\{ \frac{1890}{209} \right\} & \\\end{align}\)
9. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{8}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-18}{ \color{blue}{24} }x-\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+15& = & -18x-48 \\\Leftrightarrow & 8x \color{red}{+15} \color{blue}{-15} \color{blue}{+18x} & = & \color{red}{-18x} -48 \color{blue}{+18x} \color{blue}{-15} \\\Leftrightarrow & 26x& = & -63 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-63}{26} \\\Leftrightarrow & x = \frac{-63}{26} & & \\ & V = \left\{ \frac{-63}{26} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -70x-126 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+70x} & = & \color{red}{-70x} -126 \color{blue}{+70x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & -144 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-144}{91} \\\Leftrightarrow & x = \frac{-144}{91} & & \\ & V = \left\{ \frac{-144}{91} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 6x-126 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} -126 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & 3x& = & -130 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-130}{3} \\\Leftrightarrow & x = \frac{-130}{3} & & \\ & V = \left\{ \frac{-130}{3} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 3, 14 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{70x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 70x-45& = & 42x+420 \\\Leftrightarrow & 70x \color{red}{-45} \color{blue}{+45} \color{blue}{-42x} & = & \color{red}{42x} +420 \color{blue}{-42x} \color{blue}{+45} \\\Leftrightarrow & 28x& = & 465 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{465}{28} \\\Leftrightarrow & x = \frac{465}{28} & & \\ & V = \left\{ \frac{465}{28} \right\} & \\\end{align}\)