Vgln. eerste graad (reeks 4)

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

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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. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{5}{3}x+6 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{10}{ \color{blue}{6} }x+\frac{36}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 10x+36 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-10x} & = & \color{red}{10x} +36 \color{blue}{-10x} \color{blue}{-5} \\\Leftrightarrow & -7x& = & 31 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{31}{-7} \\\Leftrightarrow & x = \frac{-31}{7} & & \\ & V = \left\{ \frac{-31}{7} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & -130x-468 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+130x} & = & \color{red}{-130x} -468 \color{blue}{+130x} \color{blue}{-24} \\\Leftrightarrow & 169x& = & -492 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-492}{169} \\\Leftrightarrow & x = \frac{-492}{169} & & \\ & V = \left\{ \frac{-492}{169} \right\} & \\\end{align}\)
4. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 56 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{728}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+56& = & 91x-728 \\\Leftrightarrow & 26x \color{red}{+56} \color{blue}{-56} \color{blue}{-91x} & = & \color{red}{91x} -728 \color{blue}{-91x} \color{blue}{-56} \\\Leftrightarrow & -65x& = & -784 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-784}{-65} \\\Leftrightarrow & x = \frac{784}{65} & & \\ & V = \left\{ \frac{784}{65} \right\} & \\\end{align}\)
5. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 7x+70 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-7x} & = & \color{red}{7x} +70 \color{blue}{-7x} \color{blue}{-4} \\\Leftrightarrow & -5x& = & 66 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{66}{-5} \\\Leftrightarrow & x = \frac{-66}{5} & & \\ & V = \left\{ \frac{-66}{5} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & -50x-30 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{+50x} & = & \color{red}{-50x} -30 \color{blue}{+50x} \color{blue}{+4} \\\Leftrightarrow & 65x& = & -26 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{-26}{65} \\\Leftrightarrow & x = \frac{-2}{5} & & \\ & V = \left\{ \frac{-2}{5} \right\} & \\\end{align}\)
7. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{8}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }- \frac{ 15 }{ \color{blue}{40} })& = & (\frac{20}{ \color{blue}{40} }x+\frac{240}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x-15& = & 20x+240 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-20x} & = & \color{red}{20x} +240 \color{blue}{-20x} \color{blue}{+15} \\\Leftrightarrow & -12x& = & 255 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{255}{-12} \\\Leftrightarrow & x = \frac{-85}{4} & & \\ & V = \left\{ \frac{-85}{4} \right\} & \\\end{align}\)
8. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{1}{4}x-1 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 21 }{ \color{blue}{112} })& = & (\frac{28}{ \color{blue}{112} }x-\frac{112}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+21& = & 28x-112 \\\Leftrightarrow & 16x \color{red}{+21} \color{blue}{-21} \color{blue}{-28x} & = & \color{red}{28x} -112 \color{blue}{-28x} \color{blue}{-21} \\\Leftrightarrow & -12x& = & -133 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{-133}{-12} \\\Leftrightarrow & x = \frac{133}{12} & & \\ & V = \left\{ \frac{133}{12} \right\} & \\\end{align}\)
9. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x-\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & 22x-220 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-22x} & = & \color{red}{22x} -220 \color{blue}{-22x} \color{blue}{+40} \\\Leftrightarrow & 33x& = & -180 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-180}{33} \\\Leftrightarrow & x = \frac{-60}{11} & & \\ & V = \left\{ \frac{-60}{11} \right\} & \\\end{align}\)
10. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 70 }{ \color{blue}{385} })& = & (\frac{77}{ \color{blue}{385} }x+\frac{2310}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-70& = & 77x+2310 \\\Leftrightarrow & 55x \color{red}{-70} \color{blue}{+70} \color{blue}{-77x} & = & \color{red}{77x} +2310 \color{blue}{-77x} \color{blue}{+70} \\\Leftrightarrow & -22x& = & 2380 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{2380}{-22} \\\Leftrightarrow & x = \frac{-1190}{11} & & \\ & V = \left\{ \frac{-1190}{11} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 16x+240 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} +240 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & 8x& = & 225 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{225}{8} \\\Leftrightarrow & x = \frac{225}{8} & & \\ & V = \left\{ \frac{225}{8} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 40x-480 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-40x} & = & \color{red}{40x} -480 \color{blue}{-40x} \color{blue}{+15} \\\Leftrightarrow & -24x& = & -465 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-465}{-24} \\\Leftrightarrow & x = \frac{155}{8} & & \\ & V = \left\{ \frac{155}{8} \right\} & \\\end{align}\)