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

---

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

Verbetersleutel

1. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{-4}{ \color{blue}{6} }x-\frac{12}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & -4x-12 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{+4x} & = & \color{red}{-4x} -12 \color{blue}{+4x} \color{blue}{-5} \\\Leftrightarrow & 7x& = & -17 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-17}{7} \\\Leftrightarrow & x = \frac{-17}{7} & & \\ & V = \left\{ \frac{-17}{7} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{-7}{4}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & -147x-168 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{+147x} & = & \color{red}{-147x} -168 \color{blue}{+147x} \color{blue}{-48} \\\Leftrightarrow & 175x& = & -216 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-216}{175} \\\Leftrightarrow & x = \frac{-216}{175} & & \\ & V = \left\{ \frac{-216}{175} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{5}{6}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{75}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 75x-720 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-75x} & = & \color{red}{75x} -720 \color{blue}{-75x} \color{blue}{+40} \\\Leftrightarrow & -57x& = & -680 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{-680}{-57} \\\Leftrightarrow & x = \frac{680}{57} & & \\ & V = \left\{ \frac{680}{57} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -42x-90 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} -90 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 47x& = & -82 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-82}{47} \\\Leftrightarrow & x = \frac{-82}{47} & & \\ & V = \left\{ \frac{-82}{47} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & 14x-210 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-14x} & = & \color{red}{14x} -210 \color{blue}{-14x} \color{blue}{+30} \\\Leftrightarrow & 7x& = & -180 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-180}{7} \\\Leftrightarrow & x = \frac{-180}{7} & & \\ & V = \left\{ \frac{-180}{7} \right\} & \\\end{align}\)
6. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 15 }{ \color{blue}{35} })& = & (\frac{-49}{ \color{blue}{35} }x-\frac{245}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+15& = & -49x-245 \\\Leftrightarrow & 5x \color{red}{+15} \color{blue}{-15} \color{blue}{+49x} & = & \color{red}{-49x} -245 \color{blue}{+49x} \color{blue}{-15} \\\Leftrightarrow & 54x& = & -260 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{-260}{54} \\\Leftrightarrow & x = \frac{-130}{27} & & \\ & V = \left\{ \frac{-130}{27} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x-54 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} -54 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & -58 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-58}{-3} \\\Leftrightarrow & x = \frac{58}{3} & & \\ & V = \left\{ \frac{58}{3} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{4}{3}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 88x-396 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-88x} & = & \color{red}{88x} -396 \color{blue}{-88x} \color{blue}{+30} \\\Leftrightarrow & -55x& = & -366 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-366}{-55} \\\Leftrightarrow & x = \frac{366}{55} & & \\ & V = \left\{ \frac{366}{55} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{231}{ \color{blue}{66} }x-\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 231x-330 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-231x} & = & \color{red}{231x} -330 \color{blue}{-231x} \color{blue}{-30} \\\Leftrightarrow & -209x& = & -360 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-360}{-209} \\\Leftrightarrow & x = \frac{360}{209} & & \\ & V = \left\{ \frac{360}{209} \right\} & \\\end{align}\)
10. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{7}{2}x+2 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 28 }{ \color{blue}{182} })& = & (\frac{637}{ \color{blue}{182} }x+\frac{364}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-28& = & 637x+364 \\\Leftrightarrow & 26x \color{red}{-28} \color{blue}{+28} \color{blue}{-637x} & = & \color{red}{637x} +364 \color{blue}{-637x} \color{blue}{+28} \\\Leftrightarrow & -611x& = & 392 \\\Leftrightarrow & \frac{-611x}{ \color{red}{-611} }& = & \frac{392}{-611} \\\Leftrightarrow & x = \frac{-392}{611} & & \\ & V = \left\{ \frac{-392}{611} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x-90 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} -90 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & -99 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-99}{-1} \\\Leftrightarrow & x = 99 & & \\ & V = \left\{ 99 \right\} & \\\end{align}\)
12. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x+\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & -104x+468 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{+104x} & = & \color{red}{-104x} +468 \color{blue}{+104x} \color{blue}{+36} \\\Leftrightarrow & 143x& = & 504 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{504}{143} \\\Leftrightarrow & x = \frac{504}{143} & & \\ & V = \left\{ \frac{504}{143} \right\} & \\\end{align}\)