Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & -264x-2310 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{+264x} & = & \color{red}{-264x} -2310 \color{blue}{+264x} \color{blue}{-90} \\\Leftrightarrow & 319x& = & -2400 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-2400}{319} \\\Leftrightarrow & x = \frac{-2400}{319} & & \\ & V = \left\{ \frac{-2400}{319} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & 6x+126 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{-6x} & = & \color{red}{6x} +126 \color{blue}{-6x} \color{blue}{-10} \\\Leftrightarrow & 3x& = & 116 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{116}{3} \\\Leftrightarrow & x = \frac{116}{3} & & \\ & V = \left\{ \frac{116}{3} \right\} & \\\end{align}\)
3. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{8}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x-\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+9& = & 56x-72 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{-56x} & = & \color{red}{56x} -72 \color{blue}{-56x} \color{blue}{-9} \\\Leftrightarrow & -50x& = & -81 \\\Leftrightarrow & \frac{-50x}{ \color{red}{-50} }& = & \frac{-81}{-50} \\\Leftrightarrow & x = \frac{81}{50} & & \\ & V = \left\{ \frac{81}{50} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & -110x-66 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{+110x} & = & \color{red}{-110x} -66 \color{blue}{+110x} \color{blue}{+12} \\\Leftrightarrow & 143x& = & -54 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-54}{143} \\\Leftrightarrow & x = \frac{-54}{143} & & \\ & V = \left\{ \frac{-54}{143} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 6x+240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-6x} & = & \color{red}{6x} +240 \color{blue}{-6x} \color{blue}{+8} \\\Leftrightarrow & -x& = & 248 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{248}{-1} \\\Leftrightarrow & x = -248 & & \\ & V = \left\{ -248 \right\} & \\\end{align}\)
6. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 65x-910 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} -910 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -39x& = & -950 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-950}{-39} \\\Leftrightarrow & x = \frac{950}{39} & & \\ & V = \left\{ \frac{950}{39} \right\} & \\\end{align}\)
7. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+20& = & 55x+550 \\\Leftrightarrow & 22x \color{red}{+20} \color{blue}{-20} \color{blue}{-55x} & = & \color{red}{55x} +550 \color{blue}{-55x} \color{blue}{-20} \\\Leftrightarrow & -33x& = & 530 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{530}{-33} \\\Leftrightarrow & x = \frac{-530}{33} & & \\ & V = \left\{ \frac{-530}{33} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & 98x+336 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{-98x} & = & \color{red}{98x} +336 \color{blue}{-98x} \color{blue}{+9} \\\Leftrightarrow & -77x& = & 345 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{345}{-77} \\\Leftrightarrow & x = \frac{-345}{77} & & \\ & V = \left\{ \frac{-345}{77} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -36x-720 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+36x} & = & \color{red}{-36x} -720 \color{blue}{+36x} \color{blue}{+40} \\\Leftrightarrow & 51x& = & -680 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-680}{51} \\\Leftrightarrow & x = \frac{-40}{3} & & \\ & V = \left\{ \frac{-40}{3} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & -12x+90 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} +90 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 17x& = & 86 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{86}{17} \\\Leftrightarrow & x = \frac{86}{17} & & \\ & V = \left\{ \frac{86}{17} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{126}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & 126x+210 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{-126x} & = & \color{red}{126x} +210 \color{blue}{-126x} \color{blue}{+14} \\\Leftrightarrow & -111x& = & 224 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{224}{-111} \\\Leftrightarrow & x = \frac{-224}{111} & & \\ & V = \left\{ \frac{-224}{111} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & -312x+390 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+312x} & = & \color{red}{-312x} +390 \color{blue}{+312x} \color{blue}{-60} \\\Leftrightarrow & 377x& = & 330 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{330}{377} \\\Leftrightarrow & x = \frac{330}{377} & & \\ & V = \left\{ \frac{330}{377} \right\} & \\\end{align}\)