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

---

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -28x+252 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+28x} & = & \color{red}{-28x} +252 \color{blue}{+28x} \color{blue}{+18} \\\Leftrightarrow & 49x& = & 270 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{270}{49} \\\Leftrightarrow & x = \frac{270}{49} & & \\ & V = \left\{ \frac{270}{49} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -168x+210 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+168x} & = & \color{red}{-168x} +210 \color{blue}{+168x} \color{blue}{-90} \\\Leftrightarrow & 203x& = & 120 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{120}{203} \\\Leftrightarrow & x = \frac{120}{203} & & \\ & V = \left\{ \frac{120}{203} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x-126 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} -126 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -114 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-114}{-7} \\\Leftrightarrow & x = \frac{114}{7} & & \\ & V = \left\{ \frac{114}{7} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{-7}{4}x+5 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{-385}{ \color{blue}{220} }x+\frac{1100}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+100& = & -385x+1100 \\\Leftrightarrow & 44x \color{red}{+100} \color{blue}{-100} \color{blue}{+385x} & = & \color{red}{-385x} +1100 \color{blue}{+385x} \color{blue}{-100} \\\Leftrightarrow & 429x& = & 1000 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{1000}{429} \\\Leftrightarrow & x = \frac{1000}{429} & & \\ & V = \left\{ \frac{1000}{429} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{9}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{27}{ \color{blue}{18} }x+\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+10& = & 27x+144 \\\Leftrightarrow & 6x \color{red}{+10} \color{blue}{-10} \color{blue}{-27x} & = & \color{red}{27x} +144 \color{blue}{-27x} \color{blue}{-10} \\\Leftrightarrow & -21x& = & 134 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{134}{-21} \\\Leftrightarrow & x = \frac{-134}{21} & & \\ & V = \left\{ \frac{-134}{21} \right\} & \\\end{align}\)
6. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }- \frac{ 2 }{ \color{blue}{15} })& = & (\frac{35}{ \color{blue}{15} }x+\frac{30}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x-2& = & 35x+30 \\\Leftrightarrow & 3x \color{red}{-2} \color{blue}{+2} \color{blue}{-35x} & = & \color{red}{35x} +30 \color{blue}{-35x} \color{blue}{+2} \\\Leftrightarrow & -32x& = & 32 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{32}{-32} \\\Leftrightarrow & x = -1 & & \\ & V = \left\{ -1 \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-80& = & 35x+840 \\\Leftrightarrow & 28x \color{red}{-80} \color{blue}{+80} \color{blue}{-35x} & = & \color{red}{35x} +840 \color{blue}{-35x} \color{blue}{+80} \\\Leftrightarrow & -7x& = & 920 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{920}{-7} \\\Leftrightarrow & x = \frac{-920}{7} & & \\ & V = \left\{ \frac{-920}{7} \right\} & \\\end{align}\)
8. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 6 }{ \color{blue}{21} })& = & (\frac{7}{ \color{blue}{21} }x+\frac{42}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-6& = & 7x+42 \\\Leftrightarrow & 3x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} +42 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -4x& = & 48 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{48}{-4} \\\Leftrightarrow & x = -12 & & \\ & V = \left\{ -12 \right\} & \\\end{align}\)
9. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{54}{ \color{blue}{45} }x+\frac{270}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-10& = & 54x+270 \\\Leftrightarrow & 15x \color{red}{-10} \color{blue}{+10} \color{blue}{-54x} & = & \color{red}{54x} +270 \color{blue}{-54x} \color{blue}{+10} \\\Leftrightarrow & -39x& = & 280 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{280}{-39} \\\Leftrightarrow & x = \frac{-280}{39} & & \\ & V = \left\{ \frac{-280}{39} \right\} & \\\end{align}\)
10. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 4 }{ \color{blue}{15} })& = & (\frac{-21}{ \color{blue}{15} }x-\frac{105}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+4& = & -21x-105 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+21x} & = & \color{red}{-21x} -105 \color{blue}{+21x} \color{blue}{-4} \\\Leftrightarrow & 26x& = & -109 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-109}{26} \\\Leftrightarrow & x = \frac{-109}{26} & & \\ & V = \left\{ \frac{-109}{26} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 15x+150 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-15x} & = & \color{red}{15x} +150 \color{blue}{-15x} \color{blue}{-8} \\\Leftrightarrow & -9x& = & 142 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{142}{-9} \\\Leftrightarrow & x = \frac{-142}{9} & & \\ & V = \left\{ \frac{-142}{9} \right\} & \\\end{align}\)
12. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{126}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & 63x-126 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{-63x} & = & \color{red}{63x} -126 \color{blue}{-63x} \color{blue}{+70} \\\Leftrightarrow & -45x& = & -56 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-56}{-45} \\\Leftrightarrow & x = \frac{56}{45} & & \\ & V = \left\{ \frac{56}{45} \right\} & \\\end{align}\)