Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{-5}{6}x-3 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 84 }{ \color{blue}{462} })& = & (\frac{-385}{ \color{blue}{462} }x-\frac{1386}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+84& = & -385x-1386 \\\Leftrightarrow & 66x \color{red}{+84} \color{blue}{-84} \color{blue}{+385x} & = & \color{red}{-385x} -1386 \color{blue}{+385x} \color{blue}{-84} \\\Leftrightarrow & 451x& = & -1470 \\\Leftrightarrow & \frac{451x}{ \color{red}{451} }& = & \frac{-1470}{451} \\\Leftrightarrow & x = \frac{-1470}{451} & & \\ & V = \left\{ \frac{-1470}{451} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x+\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & -60x+36 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{+60x} & = & \color{red}{-60x} +36 \color{blue}{+60x} \color{blue}{-16} \\\Leftrightarrow & 69x& = & 20 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{20}{69} \\\Leftrightarrow & x = \frac{20}{69} & & \\ & V = \left\{ \frac{20}{69} \right\} & \\\end{align}\)
3. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & -462x+2640 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{+462x} & = & \color{red}{-462x} +2640 \color{blue}{+462x} \color{blue}{+120} \\\Leftrightarrow & 517x& = & 2760 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{2760}{517} \\\Leftrightarrow & x = \frac{2760}{517} & & \\ & V = \left\{ \frac{2760}{517} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 112x-252 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-112x} & = & \color{red}{112x} -252 \color{blue}{-112x} \color{blue}{-48} \\\Leftrightarrow & -91x& = & -300 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-300}{-91} \\\Leftrightarrow & x = \frac{300}{91} & & \\ & V = \left\{ \frac{300}{91} \right\} & \\\end{align}\)
5. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{65}{ \color{blue}{195} }x+\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+30& = & 65x+390 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} +390 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -26x& = & 360 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{360}{-26} \\\Leftrightarrow & x = \frac{-180}{13} & & \\ & V = \left\{ \frac{-180}{13} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & 18x+270 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{-18x} & = & \color{red}{18x} +270 \color{blue}{-18x} \color{blue}{+40} \\\Leftrightarrow & -3x& = & 310 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{310}{-3} \\\Leftrightarrow & x = \frac{-310}{3} & & \\ & V = \left\{ \frac{-310}{3} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-5}{6}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & -175x+210 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{+175x} & = & \color{red}{-175x} +210 \color{blue}{+175x} \color{blue}{-90} \\\Leftrightarrow & 217x& = & 120 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{120}{217} \\\Leftrightarrow & x = \frac{120}{217} & & \\ & V = \left\{ \frac{120}{217} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+18& = & -45x-360 \\\Leftrightarrow & 20x \color{red}{+18} \color{blue}{-18} \color{blue}{+45x} & = & \color{red}{-45x} -360 \color{blue}{+45x} \color{blue}{-18} \\\Leftrightarrow & 65x& = & -378 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{-378}{65} \\\Leftrightarrow & x = \frac{-378}{65} & & \\ & V = \left\{ \frac{-378}{65} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 5, 14 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 30 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-30& = & -105x+840 \\\Leftrightarrow & 28x \color{red}{-30} \color{blue}{+30} \color{blue}{+105x} & = & \color{red}{-105x} +840 \color{blue}{+105x} \color{blue}{+30} \\\Leftrightarrow & 133x& = & 870 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{870}{133} \\\Leftrightarrow & x = \frac{870}{133} & & \\ & V = \left\{ \frac{870}{133} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-18& = & 39x-546 \\\Leftrightarrow & 26x \color{red}{-18} \color{blue}{+18} \color{blue}{-39x} & = & \color{red}{39x} -546 \color{blue}{-39x} \color{blue}{+18} \\\Leftrightarrow & -13x& = & -528 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-528}{-13} \\\Leftrightarrow & x = \frac{528}{13} & & \\ & V = \left\{ \frac{528}{13} \right\} & \\\end{align}\)
11. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{49}{ \color{blue}{21} }x-\frac{84}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & 49x-84 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{-49x} & = & \color{red}{49x} -84 \color{blue}{-49x} \color{blue}{-9} \\\Leftrightarrow & -46x& = & -93 \\\Leftrightarrow & \frac{-46x}{ \color{red}{-46} }& = & \frac{-93}{-46} \\\Leftrightarrow & x = \frac{93}{46} & & \\ & V = \left\{ \frac{93}{46} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+50& = & 55x+220 \\\Leftrightarrow & 22x \color{red}{+50} \color{blue}{-50} \color{blue}{-55x} & = & \color{red}{55x} +220 \color{blue}{-55x} \color{blue}{-50} \\\Leftrightarrow & -33x& = & 170 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{170}{-33} \\\Leftrightarrow & x = \frac{-170}{33} & & \\ & V = \left\{ \frac{-170}{33} \right\} & \\\end{align}\)