Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{4}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 312x-1170 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-312x} & = & \color{red}{312x} -1170 \color{blue}{-312x} \color{blue}{-60} \\\Leftrightarrow & -247x& = & -1230 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-1230}{-247} \\\Leftrightarrow & x = \frac{1230}{247} & & \\ & V = \left\{ \frac{1230}{247} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{-7}{4}x+6 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-63}{ \color{blue}{36} }x+\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & -63x+216 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{+63x} & = & \color{red}{-63x} +216 \color{blue}{+63x} \color{blue}{-8} \\\Leftrightarrow & 75x& = & 208 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{208}{75} \\\Leftrightarrow & x = \frac{208}{75} & & \\ & V = \left\{ \frac{208}{75} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x-336 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} -336 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -324 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-324}{-7} \\\Leftrightarrow & x = \frac{324}{7} & & \\ & V = \left\{ \frac{324}{7} \right\} & \\\end{align}\)
4. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 30 }{ \color{blue}{165} })& = & (\frac{198}{ \color{blue}{165} }x+\frac{165}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+30& = & 198x+165 \\\Leftrightarrow & 55x \color{red}{+30} \color{blue}{-30} \color{blue}{-198x} & = & \color{red}{198x} +165 \color{blue}{-198x} \color{blue}{-30} \\\Leftrightarrow & -143x& = & 135 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{135}{-143} \\\Leftrightarrow & x = \frac{-135}{143} & & \\ & V = \left\{ \frac{-135}{143} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{105}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 105x-210 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-105x} & = & \color{red}{105x} -210 \color{blue}{-105x} \color{blue}{-20} \\\Leftrightarrow & -91x& = & -230 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-230}{-91} \\\Leftrightarrow & x = \frac{230}{91} & & \\ & V = \left\{ \frac{230}{91} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{-5}{6}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 28 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-28& = & -175x+420 \\\Leftrightarrow & 30x \color{red}{-28} \color{blue}{+28} \color{blue}{+175x} & = & \color{red}{-175x} +420 \color{blue}{+175x} \color{blue}{+28} \\\Leftrightarrow & 205x& = & 448 \\\Leftrightarrow & \frac{205x}{ \color{red}{205} }& = & \frac{448}{205} \\\Leftrightarrow & x = \frac{448}{205} & & \\ & V = \left\{ \frac{448}{205} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & 42x+210 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{-42x} & = & \color{red}{42x} +210 \color{blue}{-42x} \color{blue}{-8} \\\Leftrightarrow & -37x& = & 202 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{202}{-37} \\\Leftrightarrow & x = \frac{-202}{37} & & \\ & V = \left\{ \frac{-202}{37} \right\} & \\\end{align}\)
8. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{4}{5}x-6 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 15 }{ \color{blue}{35} })& = & (\frac{28}{ \color{blue}{35} }x-\frac{210}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+15& = & 28x-210 \\\Leftrightarrow & 5x \color{red}{+15} \color{blue}{-15} \color{blue}{-28x} & = & \color{red}{28x} -210 \color{blue}{-28x} \color{blue}{-15} \\\Leftrightarrow & -23x& = & -225 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{-225}{-23} \\\Leftrightarrow & x = \frac{225}{23} & & \\ & V = \left\{ \frac{225}{23} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 45 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x-\frac{840}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+45& = & -175x-840 \\\Leftrightarrow & 21x \color{red}{+45} \color{blue}{-45} \color{blue}{+175x} & = & \color{red}{-175x} -840 \color{blue}{+175x} \color{blue}{-45} \\\Leftrightarrow & 196x& = & -885 \\\Leftrightarrow & \frac{196x}{ \color{red}{196} }& = & \frac{-885}{196} \\\Leftrightarrow & x = \frac{-885}{196} & & \\ & V = \left\{ \frac{-885}{196} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -56x-490 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+56x} & = & \color{red}{-56x} -490 \color{blue}{+56x} \color{blue}{+20} \\\Leftrightarrow & 91x& = & -470 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-470}{91} \\\Leftrightarrow & x = \frac{-470}{91} & & \\ & V = \left\{ \frac{-470}{91} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+9& = & 21x+42 \\\Leftrightarrow & 14x \color{red}{+9} \color{blue}{-9} \color{blue}{-21x} & = & \color{red}{21x} +42 \color{blue}{-21x} \color{blue}{-9} \\\Leftrightarrow & -7x& = & 33 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{33}{-7} \\\Leftrightarrow & x = \frac{-33}{7} & & \\ & V = \left\{ \frac{-33}{7} \right\} & \\\end{align}\)
12. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x-\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & -30x-72 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{+30x} & = & \color{red}{-30x} -72 \color{blue}{+30x} \color{blue}{-10} \\\Leftrightarrow & 39x& = & -82 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-82}{39} \\\Leftrightarrow & x = \frac{-82}{39} & & \\ & V = \left\{ \frac{-82}{39} \right\} & \\\end{align}\)