Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{7}{5}x-1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{546}{ \color{blue}{390} }x-\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 546x-390 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-546x} & = & \color{red}{546x} -390 \color{blue}{-546x} \color{blue}{+60} \\\Leftrightarrow & -481x& = & -330 \\\Leftrightarrow & \frac{-481x}{ \color{red}{-481} }& = & \frac{-330}{-481} \\\Leftrightarrow & x = \frac{330}{481} & & \\ & V = \left\{ \frac{330}{481} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & -175x+1470 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{+175x} & = & \color{red}{-175x} +1470 \color{blue}{+175x} \color{blue}{-120} \\\Leftrightarrow & 217x& = & 1350 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{1350}{217} \\\Leftrightarrow & x = \frac{1350}{217} & & \\ & V = \left\{ \frac{1350}{217} \right\} & \\\end{align}\)
3. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{315.} (\frac{45x}{ \color{blue}{315} }- \frac{ 140 }{ \color{blue}{315} })& = & (\frac{-441}{ \color{blue}{315} }x+\frac{1890}{ \color{blue}{315} }) \color{blue}{.315} \\\Leftrightarrow & 45x-140& = & -441x+1890 \\\Leftrightarrow & 45x \color{red}{-140} \color{blue}{+140} \color{blue}{+441x} & = & \color{red}{-441x} +1890 \color{blue}{+441x} \color{blue}{+140} \\\Leftrightarrow & 486x& = & 2030 \\\Leftrightarrow & \frac{486x}{ \color{red}{486} }& = & \frac{2030}{486} \\\Leftrightarrow & x = \frac{1015}{243} & & \\ & V = \left\{ \frac{1015}{243} \right\} & \\\end{align}\)
4. \(\text{40 is het kleinste gemene veelvoud van 2, 8 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{8}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{40.} (\frac{20x}{ \color{blue}{40} }+ \frac{ 15 }{ \color{blue}{40} })& = & (\frac{-56}{ \color{blue}{40} }x-\frac{320}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 20x+15& = & -56x-320 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{+56x} & = & \color{red}{-56x} -320 \color{blue}{+56x} \color{blue}{-15} \\\Leftrightarrow & 76x& = & -335 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{-335}{76} \\\Leftrightarrow & x = \frac{-335}{76} & & \\ & V = \left\{ \frac{-335}{76} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-60& = & -117x-312 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{+117x} & = & \color{red}{-117x} -312 \color{blue}{+117x} \color{blue}{+60} \\\Leftrightarrow & 169x& = & -252 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-252}{169} \\\Leftrightarrow & x = \frac{-252}{169} & & \\ & V = \left\{ \frac{-252}{169} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 5x-30 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-5x} & = & \color{red}{5x} -30 \color{blue}{-5x} \color{blue}{+4} \\\Leftrightarrow & x& = & -26 \\ & V = \left\{ -26 \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-40& = & 18x+540 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{-18x} & = & \color{red}{18x} +540 \color{blue}{-18x} \color{blue}{+40} \\\Leftrightarrow & 27x& = & 580 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{580}{27} \\\Leftrightarrow & x = \frac{580}{27} & & \\ & V = \left\{ \frac{580}{27} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 56 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+56& = & 105x-420 \\\Leftrightarrow & 30x \color{red}{+56} \color{blue}{-56} \color{blue}{-105x} & = & \color{red}{105x} -420 \color{blue}{-105x} \color{blue}{-56} \\\Leftrightarrow & -75x& = & -476 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-476}{-75} \\\Leftrightarrow & x = \frac{476}{75} & & \\ & V = \left\{ \frac{476}{75} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & 98x-294 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-98x} & = & \color{red}{98x} -294 \color{blue}{-98x} \color{blue}{+30} \\\Leftrightarrow & -77x& = & -264 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-264}{-77} \\\Leftrightarrow & x = \frac{24}{7} & & \\ & V = \left\{ \frac{24}{7} \right\} & \\\end{align}\)
10. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 12x-144 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-12x} & = & \color{red}{12x} -144 \color{blue}{-12x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & -153 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-153}{-4} \\\Leftrightarrow & x = \frac{153}{4} & & \\ & V = \left\{ \frac{153}{4} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x-90 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} -90 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & -50 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-50}{-27} \\\Leftrightarrow & x = \frac{50}{27} & & \\ & V = \left\{ \frac{50}{27} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{14}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 25 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-25& = & -28x+280 \\\Leftrightarrow & 35x \color{red}{-25} \color{blue}{+25} \color{blue}{+28x} & = & \color{red}{-28x} +280 \color{blue}{+28x} \color{blue}{+25} \\\Leftrightarrow & 63x& = & 305 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{305}{63} \\\Leftrightarrow & x = \frac{305}{63} & & \\ & V = \left\{ \frac{305}{63} \right\} & \\\end{align}\)