Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{42}{ \color{blue}{18} }x-\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & 42x-90 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-42x} & = & \color{red}{42x} -90 \color{blue}{-42x} \color{blue}{-8} \\\Leftrightarrow & -33x& = & -98 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-98}{-33} \\\Leftrightarrow & x = \frac{98}{33} & & \\ & V = \left\{ \frac{98}{33} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x+\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 140x+300 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-140x} & = & \color{red}{140x} +300 \color{blue}{-140x} \color{blue}{-8} \\\Leftrightarrow & -125x& = & 292 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{292}{-125} \\\Leftrightarrow & x = \frac{-292}{125} & & \\ & V = \left\{ \frac{-292}{125} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{-5}{6}x-2 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x-\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & -75x-180 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{+75x} & = & \color{red}{-75x} -180 \color{blue}{+75x} \color{blue}{+40} \\\Leftrightarrow & 93x& = & -140 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{-140}{93} \\\Leftrightarrow & x = \frac{-140}{93} & & \\ & V = \left\{ \frac{-140}{93} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{42}{ \color{blue}{18} }x-\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & 42x-54 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{-42x} & = & \color{red}{42x} -54 \color{blue}{-42x} \color{blue}{+4} \\\Leftrightarrow & -33x& = & -50 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-50}{-33} \\\Leftrightarrow & x = \frac{50}{33} & & \\ & V = \left\{ \frac{50}{33} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 14x+252 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-14x} & = & \color{red}{14x} +252 \color{blue}{-14x} \color{blue}{+12} \\\Leftrightarrow & 7x& = & 264 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{264}{7} \\\Leftrightarrow & x = \frac{264}{7} & & \\ & V = \left\{ \frac{264}{7} \right\} & \\\end{align}\)
6. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{11}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 60 }{ \color{blue}{220} })& = & (\frac{-88}{ \color{blue}{220} }x+\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-60& = & -88x+880 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+88x} & = & \color{red}{-88x} +880 \color{blue}{+88x} \color{blue}{+60} \\\Leftrightarrow & 143x& = & 940 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{940}{143} \\\Leftrightarrow & x = \frac{940}{143} & & \\ & V = \left\{ \frac{940}{143} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & 196x+252 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-196x} & = & \color{red}{196x} +252 \color{blue}{-196x} \color{blue}{+18} \\\Leftrightarrow & -175x& = & 270 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{270}{-175} \\\Leftrightarrow & x = \frac{-54}{35} & & \\ & V = \left\{ \frac{-54}{35} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x+126 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} +126 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & 138 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{138}{-7} \\\Leftrightarrow & x = \frac{-138}{7} & & \\ & V = \left\{ \frac{-138}{7} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 45 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x-\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+45& = & 21x-525 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-21x} & = & \color{red}{21x} -525 \color{blue}{-21x} \color{blue}{-45} \\\Leftrightarrow & 14x& = & -570 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{-570}{14} \\\Leftrightarrow & x = \frac{-285}{7} & & \\ & V = \left\{ \frac{-285}{7} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 12x+180 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-12x} & = & \color{red}{12x} +180 \color{blue}{-12x} \color{blue}{-16} \\\Leftrightarrow & 3x& = & 164 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{164}{3} \\\Leftrightarrow & x = \frac{164}{3} & & \\ & V = \left\{ \frac{164}{3} \right\} & \\\end{align}\)
11. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x-\frac{24}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x+5& = & 3x-24 \\\Leftrightarrow & 2x \color{red}{+5} \color{blue}{-5} \color{blue}{-3x} & = & \color{red}{3x} -24 \color{blue}{-3x} \color{blue}{-5} \\\Leftrightarrow & -x& = & -29 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-29}{-1} \\\Leftrightarrow & x = 29 & & \\ & V = \left\{ 29 \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 28x+420 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-28x} & = & \color{red}{28x} +420 \color{blue}{-28x} \color{blue}{-80} \\\Leftrightarrow & 7x& = & 340 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{340}{7} \\\Leftrightarrow & x = \frac{340}{7} & & \\ & V = \left\{ \frac{340}{7} \right\} & \\\end{align}\)