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

---

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-80}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & -80x+30 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{+80x} & = & \color{red}{-80x} +30 \color{blue}{+80x} \color{blue}{+9} \\\Leftrightarrow & 95x& = & 39 \\\Leftrightarrow & \frac{95x}{ \color{red}{95} }& = & \frac{39}{95} \\\Leftrightarrow & x = \frac{39}{95} & & \\ & V = \left\{ \frac{39}{95} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+24& = & -63x-504 \\\Leftrightarrow & 28x \color{red}{+24} \color{blue}{-24} \color{blue}{+63x} & = & \color{red}{-63x} -504 \color{blue}{+63x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -528 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-528}{91} \\\Leftrightarrow & x = \frac{-528}{91} & & \\ & V = \left\{ \frac{-528}{91} \right\} & \\\end{align}\)
3. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{65}{ \color{blue}{195} }x-\frac{975}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+30& = & 65x-975 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} -975 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -26x& = & -1005 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-1005}{-26} \\\Leftrightarrow & x = \frac{1005}{26} & & \\ & V = \left\{ \frac{1005}{26} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{5}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 175x-210 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-175x} & = & \color{red}{175x} -210 \color{blue}{-175x} \color{blue}{-60} \\\Leftrightarrow & -133x& = & -270 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-270}{-133} \\\Leftrightarrow & x = \frac{270}{133} & & \\ & V = \left\{ \frac{270}{133} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{3}{2}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{105}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 105x+280 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-105x} & = & \color{red}{105x} +280 \color{blue}{-105x} \color{blue}{-30} \\\Leftrightarrow & -91x& = & 250 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{250}{-91} \\\Leftrightarrow & x = \frac{-250}{91} & & \\ & V = \left\{ \frac{-250}{91} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x+\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & 9x+216 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} +216 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & 3x& = & 224 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{224}{3} \\\Leftrightarrow & x = \frac{224}{3} & & \\ & V = \left\{ \frac{224}{3} \right\} & \\\end{align}\)
7. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+60& = & 275x+1980 \\\Leftrightarrow & 66x \color{red}{+60} \color{blue}{-60} \color{blue}{-275x} & = & \color{red}{275x} +1980 \color{blue}{-275x} \color{blue}{-60} \\\Leftrightarrow & -209x& = & 1920 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{1920}{-209} \\\Leftrightarrow & x = \frac{-1920}{209} & & \\ & V = \left\{ \frac{-1920}{209} \right\} & \\\end{align}\)
8. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{315.} (\frac{45x}{ \color{blue}{315} }- \frac{ 70 }{ \color{blue}{315} })& = & (\frac{63}{ \color{blue}{315} }x+\frac{315}{ \color{blue}{315} }) \color{blue}{.315} \\\Leftrightarrow & 45x-70& = & 63x+315 \\\Leftrightarrow & 45x \color{red}{-70} \color{blue}{+70} \color{blue}{-63x} & = & \color{red}{63x} +315 \color{blue}{-63x} \color{blue}{+70} \\\Leftrightarrow & -18x& = & 385 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{385}{-18} \\\Leftrightarrow & x = \frac{-385}{18} & & \\ & V = \left\{ \frac{-385}{18} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{7}{4}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+8& = & 105x+360 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-105x} & = & \color{red}{105x} +360 \color{blue}{-105x} \color{blue}{-8} \\\Leftrightarrow & -93x& = & 352 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{352}{-93} \\\Leftrightarrow & x = \frac{-352}{93} & & \\ & V = \left\{ \frac{-352}{93} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 50 }{ \color{blue}{60} })& = & (\frac{-24}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-50& = & -24x-240 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{+24x} & = & \color{red}{-24x} -240 \color{blue}{+24x} \color{blue}{+50} \\\Leftrightarrow & 39x& = & -190 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-190}{39} \\\Leftrightarrow & x = \frac{-190}{39} & & \\ & V = \left\{ \frac{-190}{39} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{2}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 84x+840 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-84x} & = & \color{red}{84x} +840 \color{blue}{-84x} \color{blue}{-120} \\\Leftrightarrow & -49x& = & 720 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{720}{-49} \\\Leftrightarrow & x = \frac{-720}{49} & & \\ & V = \left\{ \frac{-720}{49} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{3}{2}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 63x-42 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-63x} & = & \color{red}{63x} -42 \color{blue}{-63x} \color{blue}{+18} \\\Leftrightarrow & -49x& = & -24 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-24}{-49} \\\Leftrightarrow & x = \frac{24}{49} & & \\ & V = \left\{ \frac{24}{49} \right\} & \\\end{align}\)