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. \(\frac{x}{5}+\frac{3}{7}=\frac{-5}{6}x-2\)
2. \(\frac{x}{3}+\frac{4}{13}=\frac{-4}{5}x+2\)
3. \(\frac{x}{5}-\frac{4}{7}=\frac{1}{6}x+1\)
4. \(\frac{x}{5}+\frac{3}{8}=\frac{3}{2}x-7\)
5. \(\frac{x}{2}+\frac{3}{13}=\frac{-2}{5}x-7\)
6. \(\frac{x}{4}-\frac{4}{9}=\frac{2}{3}x-1\)
7. \(\frac{x}{6}-\frac{3}{14}=\frac{6}{5}x-4\)
8. \(\frac{x}{2}+\frac{4}{9}=\frac{2}{3}x+2\)
9. \(\frac{x}{4}+\frac{3}{7}=\frac{1}{3}x+2\)
10. \(\frac{x}{4}-\frac{2}{7}=\frac{7}{3}x+2\)
11. \(\frac{x}{2}-\frac{5}{16}=\frac{1}{3}x+1\)
12. \(\frac{x}{3}-\frac{4}{11}=\frac{1}{2}x-6\)

---

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-5}{6}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & -175x-420 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{+175x} & = & \color{red}{-175x} -420 \color{blue}{+175x} \color{blue}{-90} \\\Leftrightarrow & 217x& = & -510 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-510}{217} \\\Leftrightarrow & x = \frac{-510}{217} & & \\ & V = \left\{ \frac{-510}{217} \right\} & \\\end{align}\)
2. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{-156}{ \color{blue}{195} }x+\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+60& = & -156x+390 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+156x} & = & \color{red}{-156x} +390 \color{blue}{+156x} \color{blue}{-60} \\\Leftrightarrow & 221x& = & 330 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{330}{221} \\\Leftrightarrow & x = \frac{330}{221} & & \\ & V = \left\{ \frac{330}{221} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-120& = & 35x+210 \\\Leftrightarrow & 42x \color{red}{-120} \color{blue}{+120} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{+120} \\\Leftrightarrow & 7x& = & 330 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{330}{7} \\\Leftrightarrow & x = \frac{330}{7} & & \\ & V = \left\{ \frac{330}{7} \right\} & \\\end{align}\)
4. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{8}& = & \frac{3}{2}x-7 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 15 }{ \color{blue}{40} })& = & (\frac{60}{ \color{blue}{40} }x-\frac{280}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+15& = & 60x-280 \\\Leftrightarrow & 8x \color{red}{+15} \color{blue}{-15} \color{blue}{-60x} & = & \color{red}{60x} -280 \color{blue}{-60x} \color{blue}{-15} \\\Leftrightarrow & -52x& = & -295 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{-295}{-52} \\\Leftrightarrow & x = \frac{295}{52} & & \\ & V = \left\{ \frac{295}{52} \right\} & \\\end{align}\)
5. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+30& = & -52x-910 \\\Leftrightarrow & 65x \color{red}{+30} \color{blue}{-30} \color{blue}{+52x} & = & \color{red}{-52x} -910 \color{blue}{+52x} \color{blue}{-30} \\\Leftrightarrow & 117x& = & -940 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-940}{117} \\\Leftrightarrow & x = \frac{-940}{117} & & \\ & V = \left\{ \frac{-940}{117} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{2}{3}x-1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{24}{ \color{blue}{36} }x-\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & 24x-36 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{-24x} & = & \color{red}{24x} -36 \color{blue}{-24x} \color{blue}{+16} \\\Leftrightarrow & -15x& = & -20 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-20}{-15} \\\Leftrightarrow & x = \frac{4}{3} & & \\ & V = \left\{ \frac{4}{3} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & 252x-840 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{-252x} & = & \color{red}{252x} -840 \color{blue}{-252x} \color{blue}{+45} \\\Leftrightarrow & -217x& = & -795 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-795}{-217} \\\Leftrightarrow & x = \frac{795}{217} & & \\ & V = \left\{ \frac{795}{217} \right\} & \\\end{align}\)
8. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{12}{ \color{blue}{18} }x+\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & 12x+36 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-12x} & = & \color{red}{12x} +36 \color{blue}{-12x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & 28 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{28}{-3} \\\Leftrightarrow & x = \frac{-28}{3} & & \\ & V = \left\{ \frac{-28}{3} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 28x+168 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-28x} & = & \color{red}{28x} +168 \color{blue}{-28x} \color{blue}{-36} \\\Leftrightarrow & -7x& = & 132 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{132}{-7} \\\Leftrightarrow & x = \frac{-132}{7} & & \\ & V = \left\{ \frac{-132}{7} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 196x+168 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-196x} & = & \color{red}{196x} +168 \color{blue}{-196x} \color{blue}{+24} \\\Leftrightarrow & -175x& = & 192 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{192}{-175} \\\Leftrightarrow & x = \frac{-192}{175} & & \\ & V = \left\{ \frac{-192}{175} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & 16x+48 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} +48 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 8x& = & 63 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{63}{8} \\\Leftrightarrow & x = \frac{63}{8} & & \\ & V = \left\{ \frac{63}{8} \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-24& = & 33x-396 \\\Leftrightarrow & 22x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} -396 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & -372 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-372}{-11} \\\Leftrightarrow & x = \frac{372}{11} & & \\ & V = \left\{ \frac{372}{11} \right\} & \\\end{align}\)