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. \(\frac{x}{5}-\frac{4}{9}=\frac{1}{2}x+6\)
2. \(\frac{x}{7}-\frac{2}{13}=\frac{7}{4}x-3\)
3. \(\frac{x}{2}-\frac{3}{7}=\frac{1}{3}x+7\)
4. \(\frac{x}{4}+\frac{4}{15}=\frac{-5}{3}x-8\)
5. \(\frac{x}{4}-\frac{2}{7}=\frac{1}{5}x+1\)
6. \(\frac{x}{3}-\frac{2}{13}=\frac{5}{2}x-5\)
7. \(\frac{x}{2}-\frac{2}{11}=\frac{-2}{5}x-8\)
8. \(\frac{x}{7}-\frac{5}{12}=\frac{-5}{6}x-1\)
9. \(\frac{x}{5}+\frac{4}{15}=\frac{1}{6}x-1\)
10. \(\frac{x}{7}+\frac{3}{14}=\frac{-5}{3}x-8\)
11. \(\frac{x}{6}+\frac{4}{13}=\frac{-4}{5}x-7\)
12. \(\frac{x}{2}-\frac{3}{10}=\frac{-5}{3}x+3\)

---

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x+540 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} +540 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & 580 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{580}{-27} \\\Leftrightarrow & x = \frac{-580}{27} & & \\ & V = \left\{ \frac{-580}{27} \right\} & \\\end{align}\)
2. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{7}{4}x-3 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 56 }{ \color{blue}{364} })& = & (\frac{637}{ \color{blue}{364} }x-\frac{1092}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-56& = & 637x-1092 \\\Leftrightarrow & 52x \color{red}{-56} \color{blue}{+56} \color{blue}{-637x} & = & \color{red}{637x} -1092 \color{blue}{-637x} \color{blue}{+56} \\\Leftrightarrow & -585x& = & -1036 \\\Leftrightarrow & \frac{-585x}{ \color{red}{-585} }& = & \frac{-1036}{-585} \\\Leftrightarrow & x = \frac{1036}{585} & & \\ & V = \left\{ \frac{1036}{585} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 14x+294 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-14x} & = & \color{red}{14x} +294 \color{blue}{-14x} \color{blue}{+18} \\\Leftrightarrow & 7x& = & 312 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{312}{7} \\\Leftrightarrow & x = \frac{312}{7} & & \\ & V = \left\{ \frac{312}{7} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -100x-480 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+100x} & = & \color{red}{-100x} -480 \color{blue}{+100x} \color{blue}{-16} \\\Leftrightarrow & 115x& = & -496 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{-496}{115} \\\Leftrightarrow & x = \frac{-496}{115} & & \\ & V = \left\{ \frac{-496}{115} \right\} & \\\end{align}\)
5. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x+\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & 28x+140 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-28x} & = & \color{red}{28x} +140 \color{blue}{-28x} \color{blue}{+40} \\\Leftrightarrow & 7x& = & 180 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{180}{7} \\\Leftrightarrow & x = \frac{180}{7} & & \\ & V = \left\{ \frac{180}{7} \right\} & \\\end{align}\)
6. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{195}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 195x-390 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-195x} & = & \color{red}{195x} -390 \color{blue}{-195x} \color{blue}{+12} \\\Leftrightarrow & -169x& = & -378 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-378}{-169} \\\Leftrightarrow & x = \frac{378}{169} & & \\ & V = \left\{ \frac{378}{169} \right\} & \\\end{align}\)
7. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{-44}{ \color{blue}{110} }x-\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-20& = & -44x-880 \\\Leftrightarrow & 55x \color{red}{-20} \color{blue}{+20} \color{blue}{+44x} & = & \color{red}{-44x} -880 \color{blue}{+44x} \color{blue}{+20} \\\Leftrightarrow & 99x& = & -860 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{-860}{99} \\\Leftrightarrow & x = \frac{-860}{99} & & \\ & V = \left\{ \frac{-860}{99} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{-5}{6}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{-70}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & -70x-84 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{+70x} & = & \color{red}{-70x} -84 \color{blue}{+70x} \color{blue}{+35} \\\Leftrightarrow & 82x& = & -49 \\\Leftrightarrow & \frac{82x}{ \color{red}{82} }& = & \frac{-49}{82} \\\Leftrightarrow & x = \frac{-49}{82} & & \\ & V = \left\{ \frac{-49}{82} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 5x-30 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-5x} & = & \color{red}{5x} -30 \color{blue}{-5x} \color{blue}{-8} \\\Leftrightarrow & x& = & -38 \\ & V = \left\{ -38 \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+9& = & -70x-336 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{+70x} & = & \color{red}{-70x} -336 \color{blue}{+70x} \color{blue}{-9} \\\Leftrightarrow & 76x& = & -345 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{-345}{76} \\\Leftrightarrow & x = \frac{-345}{76} & & \\ & V = \left\{ \frac{-345}{76} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -312x-2730 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+312x} & = & \color{red}{-312x} -2730 \color{blue}{+312x} \color{blue}{-120} \\\Leftrightarrow & 377x& = & -2850 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-2850}{377} \\\Leftrightarrow & x = \frac{-2850}{377} & & \\ & V = \left\{ \frac{-2850}{377} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & -50x+90 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{+50x} & = & \color{red}{-50x} +90 \color{blue}{+50x} \color{blue}{+9} \\\Leftrightarrow & 65x& = & 99 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{99}{65} \\\Leftrightarrow & x = \frac{99}{65} & & \\ & V = \left\{ \frac{99}{65} \right\} & \\\end{align}\)