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

---

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{455}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+20& = & 455x-910 \\\Leftrightarrow & 26x \color{red}{+20} \color{blue}{-20} \color{blue}{-455x} & = & \color{red}{455x} -910 \color{blue}{-455x} \color{blue}{-20} \\\Leftrightarrow & -429x& = & -930 \\\Leftrightarrow & \frac{-429x}{ \color{red}{-429} }& = & \frac{-930}{-429} \\\Leftrightarrow & x = \frac{310}{143} & & \\ & V = \left\{ \frac{310}{143} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -70x-42 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+70x} & = & \color{red}{-70x} -42 \color{blue}{+70x} \color{blue}{+12} \\\Leftrightarrow & 91x& = & -30 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-30}{91} \\\Leftrightarrow & x = \frac{-30}{91} & & \\ & V = \left\{ \frac{-30}{91} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 6x-30 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} -30 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & -x& = & -34 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-34}{-1} \\\Leftrightarrow & x = 34 & & \\ & V = \left\{ 34 \right\} & \\\end{align}\)
4. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 288x+1200 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-288x} & = & \color{red}{288x} +1200 \color{blue}{-288x} \color{blue}{-45} \\\Leftrightarrow & -248x& = & 1155 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{1155}{-248} \\\Leftrightarrow & x = \frac{-1155}{248} & & \\ & V = \left\{ \frac{-1155}{248} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{200}{ \color{blue}{80} }x-\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 200x-560 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-200x} & = & \color{red}{200x} -560 \color{blue}{-200x} \color{blue}{+15} \\\Leftrightarrow & -184x& = & -545 \\\Leftrightarrow & \frac{-184x}{ \color{red}{-184} }& = & \frac{-545}{-184} \\\Leftrightarrow & x = \frac{545}{184} & & \\ & V = \left\{ \frac{545}{184} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & 84x-280 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-84x} & = & \color{red}{84x} -280 \color{blue}{-84x} \color{blue}{+40} \\\Leftrightarrow & -49x& = & -240 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-240}{-49} \\\Leftrightarrow & x = \frac{240}{49} & & \\ & V = \left\{ \frac{240}{49} \right\} & \\\end{align}\)
7. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-18}{ \color{blue}{24} }x-\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & -18x-72 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{+18x} & = & \color{red}{-18x} -72 \color{blue}{+18x} \color{blue}{-9} \\\Leftrightarrow & 26x& = & -81 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-81}{26} \\\Leftrightarrow & x = \frac{-81}{26} & & \\ & V = \left\{ \frac{-81}{26} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-80}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -80x+150 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+80x} & = & \color{red}{-80x} +150 \color{blue}{+80x} \color{blue}{+8} \\\Leftrightarrow & 95x& = & 158 \\\Leftrightarrow & \frac{95x}{ \color{red}{95} }& = & \frac{158}{95} \\\Leftrightarrow & x = \frac{158}{95} & & \\ & V = \left\{ \frac{158}{95} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & 66x-1650 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{-66x} & = & \color{red}{66x} -1650 \color{blue}{-66x} \color{blue}{-90} \\\Leftrightarrow & -11x& = & -1740 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-1740}{-11} \\\Leftrightarrow & x = \frac{1740}{11} & & \\ & V = \left\{ \frac{1740}{11} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & 14x-490 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{-14x} & = & \color{red}{14x} -490 \color{blue}{-14x} \color{blue}{+30} \\\Leftrightarrow & 21x& = & -460 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-460}{21} \\\Leftrightarrow & x = \frac{-460}{21} & & \\ & V = \left\{ \frac{-460}{21} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & -44x-264 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+44x} & = & \color{red}{-44x} -264 \color{blue}{+44x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & -240 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-240}{77} \\\Leftrightarrow & x = \frac{-240}{77} & & \\ & V = \left\{ \frac{-240}{77} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{3}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 234x-1170 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-234x} & = & \color{red}{234x} -1170 \color{blue}{-234x} \color{blue}{-60} \\\Leftrightarrow & -169x& = & -1230 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-1230}{-169} \\\Leftrightarrow & x = \frac{1230}{169} & & \\ & V = \left\{ \frac{1230}{169} \right\} & \\\end{align}\)