Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x+\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & 88x+462 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-88x} & = & \color{red}{88x} +462 \color{blue}{-88x} \color{blue}{-24} \\\Leftrightarrow & -55x& = & 438 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{438}{-55} \\\Leftrightarrow & x = \frac{-438}{55} & & \\ & V = \left\{ \frac{-438}{55} \right\} & \\\end{align}\)
2. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{-84}{ \color{blue}{112} }x-\frac{784}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & -84x-784 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{+84x} & = & \color{red}{-84x} -784 \color{blue}{+84x} \color{blue}{+35} \\\Leftrightarrow & 100x& = & -749 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{-749}{100} \\\Leftrightarrow & x = \frac{-749}{100} & & \\ & V = \left\{ \frac{-749}{100} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x-\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & -12x-108 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+12x} & = & \color{red}{-12x} -108 \color{blue}{+12x} \color{blue}{-8} \\\Leftrightarrow & 21x& = & -116 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-116}{21} \\\Leftrightarrow & x = \frac{-116}{21} & & \\ & V = \left\{ \frac{-116}{21} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x+252 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} +252 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & 240 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{240}{-7} \\\Leftrightarrow & x = \frac{-240}{7} & & \\ & V = \left\{ \frac{-240}{7} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -56x+588 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+56x} & = & \color{red}{-56x} +588 \color{blue}{+56x} \color{blue}{-48} \\\Leftrightarrow & 77x& = & 540 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{540}{77} \\\Leftrightarrow & x = \frac{540}{77} & & \\ & V = \left\{ \frac{540}{77} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x+210 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} +210 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & 270 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{270}{-7} \\\Leftrightarrow & x = \frac{-270}{7} & & \\ & V = \left\{ \frac{-270}{7} \right\} & \\\end{align}\)
7. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x-\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & 55x-1320 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{-55x} & = & \color{red}{55x} -1320 \color{blue}{-55x} \color{blue}{-120} \\\Leftrightarrow & 11x& = & -1440 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-1440}{11} \\\Leftrightarrow & x = \frac{-1440}{11} & & \\ & V = \left\{ \frac{-1440}{11} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 16x+48 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} +48 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & 8x& = & 33 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{33}{8} \\\Leftrightarrow & x = \frac{33}{8} & & \\ & V = \left\{ \frac{33}{8} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{165}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 165x-264 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-165x} & = & \color{red}{165x} -264 \color{blue}{-165x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & -288 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-288}{-143} \\\Leftrightarrow & x = \frac{288}{143} & & \\ & V = \left\{ \frac{288}{143} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & -462x+2640 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{+462x} & = & \color{red}{-462x} +2640 \color{blue}{+462x} \color{blue}{+120} \\\Leftrightarrow & 517x& = & 2760 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{2760}{517} \\\Leftrightarrow & x = \frac{2760}{517} & & \\ & V = \left\{ \frac{2760}{517} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -36x+360 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+36x} & = & \color{red}{-36x} +360 \color{blue}{+36x} \color{blue}{-20} \\\Leftrightarrow & 51x& = & 340 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{340}{51} \\\Leftrightarrow & x = \frac{20}{3} & & \\ & V = \left\{ \frac{20}{3} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -220x+132 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+220x} & = & \color{red}{-220x} +132 \color{blue}{+220x} \color{blue}{-24} \\\Leftrightarrow & 253x& = & 108 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{108}{253} \\\Leftrightarrow & x = \frac{108}{253} & & \\ & V = \left\{ \frac{108}{253} \right\} & \\\end{align}\)