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

---

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{7}{4}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{147}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+36& = & 147x+588 \\\Leftrightarrow & 28x \color{red}{+36} \color{blue}{-36} \color{blue}{-147x} & = & \color{red}{147x} +588 \color{blue}{-147x} \color{blue}{-36} \\\Leftrightarrow & -119x& = & 552 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{552}{-119} \\\Leftrightarrow & x = \frac{-552}{119} & & \\ & V = \left\{ \frac{-552}{119} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -126x-360 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+126x} & = & \color{red}{-126x} -360 \color{blue}{+126x} \color{blue}{+40} \\\Leftrightarrow & 141x& = & -320 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-320}{141} \\\Leftrightarrow & x = \frac{-320}{141} & & \\ & V = \left\{ \frac{-320}{141} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & -50x+90 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{+50x} & = & \color{red}{-50x} +90 \color{blue}{+50x} \color{blue}{-9} \\\Leftrightarrow & 65x& = & 81 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{81}{65} \\\Leftrightarrow & x = \frac{81}{65} & & \\ & V = \left\{ \frac{81}{65} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & 14x-140 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-14x} & = & \color{red}{14x} -140 \color{blue}{-14x} \color{blue}{-30} \\\Leftrightarrow & 21x& = & -170 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-170}{21} \\\Leftrightarrow & x = \frac{-170}{21} & & \\ & V = \left\{ \frac{-170}{21} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{5}{4}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-16& = & 75x-420 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{-75x} & = & \color{red}{75x} -420 \color{blue}{-75x} \color{blue}{+16} \\\Leftrightarrow & -63x& = & -404 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{-404}{-63} \\\Leftrightarrow & x = \frac{404}{63} & & \\ & V = \left\{ \frac{404}{63} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{11}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-90& = & 66x+330 \\\Leftrightarrow & 55x \color{red}{-90} \color{blue}{+90} \color{blue}{-66x} & = & \color{red}{66x} +330 \color{blue}{-66x} \color{blue}{+90} \\\Leftrightarrow & -11x& = & 420 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{420}{-11} \\\Leftrightarrow & x = \frac{-420}{11} & & \\ & V = \left\{ \frac{-420}{11} \right\} & \\\end{align}\)
7. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 12 }{ \color{blue}{21} })& = & (\frac{-35}{ \color{blue}{21} }x-\frac{42}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+12& = & -35x-42 \\\Leftrightarrow & 3x \color{red}{+12} \color{blue}{-12} \color{blue}{+35x} & = & \color{red}{-35x} -42 \color{blue}{+35x} \color{blue}{-12} \\\Leftrightarrow & 38x& = & -54 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{-54}{38} \\\Leftrightarrow & x = \frac{-27}{19} & & \\ & V = \left\{ \frac{-27}{19} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-80& = & 35x+840 \\\Leftrightarrow & 28x \color{red}{-80} \color{blue}{+80} \color{blue}{-35x} & = & \color{red}{35x} +840 \color{blue}{-35x} \color{blue}{+80} \\\Leftrightarrow & -7x& = & 920 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{920}{-7} \\\Leftrightarrow & x = \frac{-920}{7} & & \\ & V = \left\{ \frac{-920}{7} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{144}{ \color{blue}{120} }x-\frac{120}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & 144x-120 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{-144x} & = & \color{red}{144x} -120 \color{blue}{-144x} \color{blue}{-45} \\\Leftrightarrow & -124x& = & -165 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{-165}{-124} \\\Leftrightarrow & x = \frac{165}{124} & & \\ & V = \left\{ \frac{165}{124} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & -40x-180 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+40x} & = & \color{red}{-40x} -180 \color{blue}{+40x} \color{blue}{+8} \\\Leftrightarrow & 55x& = & -172 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{-172}{55} \\\Leftrightarrow & x = \frac{-172}{55} & & \\ & V = \left\{ \frac{-172}{55} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 24x+48 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-24x} & = & \color{red}{24x} +48 \color{blue}{-24x} \color{blue}{+9} \\\Leftrightarrow & -8x& = & 57 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{57}{-8} \\\Leftrightarrow & x = \frac{-57}{8} & & \\ & V = \left\{ \frac{-57}{8} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{14}& = & \frac{2}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 30 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-30& = & 56x+336 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-56x} & = & \color{red}{56x} +336 \color{blue}{-56x} \color{blue}{+30} \\\Leftrightarrow & -35x& = & 366 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{366}{-35} \\\Leftrightarrow & x = \frac{-366}{35} & & \\ & V = \left\{ \frac{-366}{35} \right\} & \\\end{align}\)