Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -126x+720 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+126x} & = & \color{red}{-126x} +720 \color{blue}{+126x} \color{blue}{-40} \\\Leftrightarrow & 141x& = & 680 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{680}{141} \\\Leftrightarrow & x = \frac{680}{141} & & \\ & V = \left\{ \frac{680}{141} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 15x-180 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-15x} & = & \color{red}{15x} -180 \color{blue}{-15x} \color{blue}{-8} \\\Leftrightarrow & -9x& = & -188 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-188}{-9} \\\Leftrightarrow & x = \frac{188}{9} & & \\ & V = \left\{ \frac{188}{9} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-2}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -84x+1260 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} +1260 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & 1200 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{1200}{119} \\\Leftrightarrow & x = \frac{1200}{119} & & \\ & V = \left\{ \frac{1200}{119} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{14}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+9& = & 14x-210 \\\Leftrightarrow & 21x \color{red}{+9} \color{blue}{-9} \color{blue}{-14x} & = & \color{red}{14x} -210 \color{blue}{-14x} \color{blue}{-9} \\\Leftrightarrow & 7x& = & -219 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-219}{7} \\\Leftrightarrow & x = \frac{-219}{7} & & \\ & V = \left\{ \frac{-219}{7} \right\} & \\\end{align}\)
5. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 105 }{ \color{blue}{231} })& = & (\frac{77}{ \color{blue}{231} }x+\frac{1617}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-105& = & 77x+1617 \\\Leftrightarrow & 33x \color{red}{-105} \color{blue}{+105} \color{blue}{-77x} & = & \color{red}{77x} +1617 \color{blue}{-77x} \color{blue}{+105} \\\Leftrightarrow & -44x& = & 1722 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{1722}{-44} \\\Leftrightarrow & x = \frac{-861}{22} & & \\ & V = \left\{ \frac{-861}{22} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{7}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & 42x-30 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{-42x} & = & \color{red}{42x} -30 \color{blue}{-42x} \color{blue}{-8} \\\Leftrightarrow & -37x& = & -38 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{-38}{-37} \\\Leftrightarrow & x = \frac{38}{37} & & \\ & V = \left\{ \frac{38}{37} \right\} & \\\end{align}\)
7. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{88}{ \color{blue}{220} }x-\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & 88x-880 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-88x} & = & \color{red}{88x} -880 \color{blue}{-88x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & -840 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-840}{-33} \\\Leftrightarrow & x = \frac{280}{11} & & \\ & V = \left\{ \frac{280}{11} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 50 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-50& = & 14x-490 \\\Leftrightarrow & 35x \color{red}{-50} \color{blue}{+50} \color{blue}{-14x} & = & \color{red}{14x} -490 \color{blue}{-14x} \color{blue}{+50} \\\Leftrightarrow & 21x& = & -440 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-440}{21} \\\Leftrightarrow & x = \frac{-440}{21} & & \\ & V = \left\{ \frac{-440}{21} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{7}{4}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+8& = & 105x-300 \\\Leftrightarrow & 20x \color{red}{+8} \color{blue}{-8} \color{blue}{-105x} & = & \color{red}{105x} -300 \color{blue}{-105x} \color{blue}{-8} \\\Leftrightarrow & -85x& = & -308 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-308}{-85} \\\Leftrightarrow & x = \frac{308}{85} & & \\ & V = \left\{ \frac{308}{85} \right\} & \\\end{align}\)
10. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{-7}{4}x-6 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 8 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x-\frac{168}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+8& = & -49x-168 \\\Leftrightarrow & 4x \color{red}{+8} \color{blue}{-8} \color{blue}{+49x} & = & \color{red}{-49x} -168 \color{blue}{+49x} \color{blue}{-8} \\\Leftrightarrow & 53x& = & -176 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{-176}{53} \\\Leftrightarrow & x = \frac{-176}{53} & & \\ & V = \left\{ \frac{-176}{53} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+25& = & -42x-180 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{+42x} & = & \color{red}{-42x} -180 \color{blue}{+42x} \color{blue}{-25} \\\Leftrightarrow & 57x& = & -205 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-205}{57} \\\Leftrightarrow & x = \frac{-205}{57} & & \\ & V = \left\{ \frac{-205}{57} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{3}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 126x+630 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-126x} & = & \color{red}{126x} +630 \color{blue}{-126x} \color{blue}{-150} \\\Leftrightarrow & -91x& = & 480 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{480}{-91} \\\Leftrightarrow & x = \frac{-480}{91} & & \\ & V = \left\{ \frac{-480}{91} \right\} & \\\end{align}\)