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. \(\frac{x}{3}+\frac{4}{15}=\frac{1}{2}x-1\)
2. \(\frac{x}{3}-\frac{3}{13}=\frac{-3}{4}x+6\)
3. \(\frac{x}{7}-\frac{3}{7}=\frac{-7}{5}x-5\)
4. \(\frac{x}{2}+\frac{3}{7}=\frac{-2}{3}x+3\)
5. \(\frac{x}{3}+\frac{4}{7}=\frac{1}{2}x-5\)
6. \(\frac{x}{4}-\frac{5}{11}=\frac{6}{5}x-2\)
7. \(\frac{x}{3}+\frac{4}{9}=\frac{7}{2}x-7\)
8. \(\frac{x}{3}+\frac{2}{7}=\frac{1}{2}x+8\)
9. \(\frac{x}{3}-\frac{3}{10}=\frac{1}{2}x+4\)
10. \(\frac{x}{7}+\frac{5}{16}=\frac{1}{4}x+5\)
11. \(\frac{x}{3}+\frac{3}{14}=\frac{6}{5}x+3\)
12. \(\frac{x}{2}+\frac{4}{7}=\frac{-4}{5}x+7\)

---

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+8& = & 15x-30 \\\Leftrightarrow & 10x \color{red}{+8} \color{blue}{-8} \color{blue}{-15x} & = & \color{red}{15x} -30 \color{blue}{-15x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & -38 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-38}{-5} \\\Leftrightarrow & x = \frac{38}{5} & & \\ & V = \left\{ \frac{38}{5} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & -117x+936 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{+117x} & = & \color{red}{-117x} +936 \color{blue}{+117x} \color{blue}{+36} \\\Leftrightarrow & 169x& = & 972 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{972}{169} \\\Leftrightarrow & x = \frac{972}{169} & & \\ & V = \left\{ \frac{972}{169} \right\} & \\\end{align}\)
3. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 15 }{ \color{blue}{35} })& = & (\frac{-49}{ \color{blue}{35} }x-\frac{175}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-15& = & -49x-175 \\\Leftrightarrow & 5x \color{red}{-15} \color{blue}{+15} \color{blue}{+49x} & = & \color{red}{-49x} -175 \color{blue}{+49x} \color{blue}{+15} \\\Leftrightarrow & 54x& = & -160 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{-160}{54} \\\Leftrightarrow & x = \frac{-80}{27} & & \\ & V = \left\{ \frac{-80}{27} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -28x+126 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+28x} & = & \color{red}{-28x} +126 \color{blue}{+28x} \color{blue}{-18} \\\Leftrightarrow & 49x& = & 108 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{108}{49} \\\Leftrightarrow & x = \frac{108}{49} & & \\ & V = \left\{ \frac{108}{49} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-210 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -234 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-234}{-7} \\\Leftrightarrow & x = \frac{234}{7} & & \\ & V = \left\{ \frac{234}{7} \right\} & \\\end{align}\)
6. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 100 }{ \color{blue}{220} })& = & (\frac{264}{ \color{blue}{220} }x-\frac{440}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-100& = & 264x-440 \\\Leftrightarrow & 55x \color{red}{-100} \color{blue}{+100} \color{blue}{-264x} & = & \color{red}{264x} -440 \color{blue}{-264x} \color{blue}{+100} \\\Leftrightarrow & -209x& = & -340 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-340}{-209} \\\Leftrightarrow & x = \frac{340}{209} & & \\ & V = \left\{ \frac{340}{209} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{63}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 63x-126 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-63x} & = & \color{red}{63x} -126 \color{blue}{-63x} \color{blue}{-8} \\\Leftrightarrow & -57x& = & -134 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{-134}{-57} \\\Leftrightarrow & x = \frac{134}{57} & & \\ & V = \left\{ \frac{134}{57} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x+336 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} +336 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & 324 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{324}{-7} \\\Leftrightarrow & x = \frac{-324}{7} & & \\ & V = \left\{ \frac{-324}{7} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 15x+120 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-15x} & = & \color{red}{15x} +120 \color{blue}{-15x} \color{blue}{+9} \\\Leftrightarrow & -5x& = & 129 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{129}{-5} \\\Leftrightarrow & x = \frac{-129}{5} & & \\ & V = \left\{ \frac{-129}{5} \right\} & \\\end{align}\)
10. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{28}{ \color{blue}{112} }x+\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 28x+560 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-28x} & = & \color{red}{28x} +560 \color{blue}{-28x} \color{blue}{-35} \\\Leftrightarrow & -12x& = & 525 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{525}{-12} \\\Leftrightarrow & x = \frac{-175}{4} & & \\ & V = \left\{ \frac{-175}{4} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 3, 14 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{70x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 70x+45& = & 252x+630 \\\Leftrightarrow & 70x \color{red}{+45} \color{blue}{-45} \color{blue}{-252x} & = & \color{red}{252x} +630 \color{blue}{-252x} \color{blue}{-45} \\\Leftrightarrow & -182x& = & 585 \\\Leftrightarrow & \frac{-182x}{ \color{red}{-182} }& = & \frac{585}{-182} \\\Leftrightarrow & x = \frac{-45}{14} & & \\ & V = \left\{ \frac{-45}{14} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & -56x+490 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+56x} & = & \color{red}{-56x} +490 \color{blue}{+56x} \color{blue}{-40} \\\Leftrightarrow & 91x& = & 450 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{450}{91} \\\Leftrightarrow & x = \frac{450}{91} & & \\ & V = \left\{ \frac{450}{91} \right\} & \\\end{align}\)