Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{-5}{6}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & -275x-1980 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{+275x} & = & \color{red}{-275x} -1980 \color{blue}{+275x} \color{blue}{-120} \\\Leftrightarrow & 341x& = & -2100 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{-2100}{341} \\\Leftrightarrow & x = \frac{-2100}{341} & & \\ & V = \left\{ \frac{-2100}{341} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & 98x+294 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{-98x} & = & \color{red}{98x} +294 \color{blue}{-98x} \color{blue}{-18} \\\Leftrightarrow & -77x& = & 276 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{276}{-77} \\\Leftrightarrow & x = \frac{-276}{77} & & \\ & V = \left\{ \frac{-276}{77} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & 56x+294 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{-56x} & = & \color{red}{56x} +294 \color{blue}{-56x} \color{blue}{+9} \\\Leftrightarrow & -35x& = & 303 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{303}{-35} \\\Leftrightarrow & x = \frac{-303}{35} & & \\ & V = \left\{ \frac{-303}{35} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{7}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 100 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-100& = & 28x+420 \\\Leftrightarrow & 35x \color{red}{-100} \color{blue}{+100} \color{blue}{-28x} & = & \color{red}{28x} +420 \color{blue}{-28x} \color{blue}{+100} \\\Leftrightarrow & 7x& = & 520 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{520}{7} \\\Leftrightarrow & x = \frac{520}{7} & & \\ & V = \left\{ \frac{520}{7} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 5, 6 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{5}{4}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 50 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-50& = & 75x+180 \\\Leftrightarrow & 12x \color{red}{-50} \color{blue}{+50} \color{blue}{-75x} & = & \color{red}{75x} +180 \color{blue}{-75x} \color{blue}{+50} \\\Leftrightarrow & -63x& = & 230 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{230}{-63} \\\Leftrightarrow & x = \frac{-230}{63} & & \\ & V = \left\{ \frac{-230}{63} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{100}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 100x+420 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-100x} & = & \color{red}{100x} +420 \color{blue}{-100x} \color{blue}{+16} \\\Leftrightarrow & -85x& = & 436 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{436}{-85} \\\Leftrightarrow & x = \frac{-436}{85} & & \\ & V = \left\{ \frac{-436}{85} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{3}{2}x+3 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{117}{ \color{blue}{78} }x+\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 117x+234 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-117x} & = & \color{red}{117x} +234 \color{blue}{-117x} \color{blue}{-12} \\\Leftrightarrow & -91x& = & 222 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{222}{-91} \\\Leftrightarrow & x = \frac{-222}{91} & & \\ & V = \left\{ \frac{-222}{91} \right\} & \\\end{align}\)
8. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 105 }{ \color{blue}{231} })& = & (\frac{77}{ \color{blue}{231} }x+\frac{462}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-105& = & 77x+462 \\\Leftrightarrow & 33x \color{red}{-105} \color{blue}{+105} \color{blue}{-77x} & = & \color{red}{77x} +462 \color{blue}{-77x} \color{blue}{+105} \\\Leftrightarrow & -44x& = & 567 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{567}{-44} \\\Leftrightarrow & x = \frac{-567}{44} & & \\ & V = \left\{ \frac{-567}{44} \right\} & \\\end{align}\)
9. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x-\frac{825}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-75& = & 55x-825 \\\Leftrightarrow & 33x \color{red}{-75} \color{blue}{+75} \color{blue}{-55x} & = & \color{red}{55x} -825 \color{blue}{-55x} \color{blue}{+75} \\\Leftrightarrow & -22x& = & -750 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{-750}{-22} \\\Leftrightarrow & x = \frac{375}{11} & & \\ & V = \left\{ \frac{375}{11} \right\} & \\\end{align}\)
10. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{165}{ \color{blue}{110} }x+\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+20& = & 165x+550 \\\Leftrightarrow & 22x \color{red}{+20} \color{blue}{-20} \color{blue}{-165x} & = & \color{red}{165x} +550 \color{blue}{-165x} \color{blue}{-20} \\\Leftrightarrow & -143x& = & 530 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{530}{-143} \\\Leftrightarrow & x = \frac{-530}{143} & & \\ & V = \left\{ \frac{-530}{143} \right\} & \\\end{align}\)
11. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 6 }{ \color{blue}{21} })& = & (\frac{-14}{ \color{blue}{21} }x+\frac{126}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-6& = & -14x+126 \\\Leftrightarrow & 3x \color{red}{-6} \color{blue}{+6} \color{blue}{+14x} & = & \color{red}{-14x} +126 \color{blue}{+14x} \color{blue}{+6} \\\Leftrightarrow & 17x& = & 132 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{132}{17} \\\Leftrightarrow & x = \frac{132}{17} & & \\ & V = \left\{ \frac{132}{17} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 105x+1050 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-105x} & = & \color{red}{105x} +1050 \color{blue}{-105x} \color{blue}{-28} \\\Leftrightarrow & -75x& = & 1022 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{1022}{-75} \\\Leftrightarrow & x = \frac{-1022}{75} & & \\ & V = \left\{ \frac{-1022}{75} \right\} & \\\end{align}\)