Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-15& = & -80x-96 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+80x} & = & \color{red}{-80x} -96 \color{blue}{+80x} \color{blue}{+15} \\\Leftrightarrow & 92x& = & -81 \\\Leftrightarrow & \frac{92x}{ \color{red}{92} }& = & \frac{-81}{92} \\\Leftrightarrow & x = \frac{-81}{92} & & \\ & V = \left\{ \frac{-81}{92} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-25& = & 5x-240 \\\Leftrightarrow & 6x \color{red}{-25} \color{blue}{+25} \color{blue}{-5x} & = & \color{red}{5x} -240 \color{blue}{-5x} \color{blue}{+25} \\\Leftrightarrow & x& = & -215 \\ & V = \left\{ -215 \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x-\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-10& = & -12x-54 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{+12x} & = & \color{red}{-12x} -54 \color{blue}{+12x} \color{blue}{+10} \\\Leftrightarrow & 21x& = & -44 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-44}{21} \\\Leftrightarrow & x = \frac{-44}{21} & & \\ & V = \left\{ \frac{-44}{21} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 75x-150 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-75x} & = & \color{red}{75x} -150 \color{blue}{-75x} \color{blue}{+4} \\\Leftrightarrow & -65x& = & -146 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-146}{-65} \\\Leftrightarrow & x = \frac{146}{65} & & \\ & V = \left\{ \frac{146}{65} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-60& = & 44x-528 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{-44x} & = & \color{red}{44x} -528 \color{blue}{-44x} \color{blue}{+60} \\\Leftrightarrow & -11x& = & -468 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-468}{-11} \\\Leftrightarrow & x = \frac{468}{11} & & \\ & V = \left\{ \frac{468}{11} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -36x+270 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+36x} & = & \color{red}{-36x} +270 \color{blue}{+36x} \color{blue}{-20} \\\Leftrightarrow & 51x& = & 250 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{250}{51} \\\Leftrightarrow & x = \frac{250}{51} & & \\ & V = \left\{ \frac{250}{51} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & -56x+140 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{+56x} & = & \color{red}{-56x} +140 \color{blue}{+56x} \color{blue}{+30} \\\Leftrightarrow & 91x& = & 170 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{170}{91} \\\Leftrightarrow & x = \frac{170}{91} & & \\ & V = \left\{ \frac{170}{91} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x-\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & -99x-132 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{+99x} & = & \color{red}{-99x} -132 \color{blue}{+99x} \color{blue}{+48} \\\Leftrightarrow & 143x& = & -84 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-84}{143} \\\Leftrightarrow & x = \frac{-84}{143} & & \\ & V = \left\{ \frac{-84}{143} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -168x-1260 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+168x} & = & \color{red}{-168x} -1260 \color{blue}{+168x} \color{blue}{-90} \\\Leftrightarrow & 203x& = & -1350 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1350}{203} \\\Leftrightarrow & x = \frac{-1350}{203} & & \\ & V = \left\{ \frac{-1350}{203} \right\} & \\\end{align}\)
10. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{4}{5}x-2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{28}{ \color{blue}{35} }x-\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 28x-70 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-28x} & = & \color{red}{28x} -70 \color{blue}{-28x} \color{blue}{-20} \\\Leftrightarrow & -23x& = & -90 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{-90}{-23} \\\Leftrightarrow & x = \frac{90}{23} & & \\ & V = \left\{ \frac{90}{23} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-120& = & 35x+1050 \\\Leftrightarrow & 42x \color{red}{-120} \color{blue}{+120} \color{blue}{-35x} & = & \color{red}{35x} +1050 \color{blue}{-35x} \color{blue}{+120} \\\Leftrightarrow & 7x& = & 1170 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{1170}{7} \\\Leftrightarrow & x = \frac{1170}{7} & & \\ & V = \left\{ \frac{1170}{7} \right\} & \\\end{align}\)
12. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & -260x-780 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{+260x} & = & \color{red}{-260x} -780 \color{blue}{+260x} \color{blue}{-36} \\\Leftrightarrow & 299x& = & -816 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-816}{299} \\\Leftrightarrow & x = \frac{-816}{299} & & \\ & V = \left\{ \frac{-816}{299} \right\} & \\\end{align}\)