Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+30& = & 56x-168 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-56x} & = & \color{red}{56x} -168 \color{blue}{-56x} \color{blue}{-30} \\\Leftrightarrow & -35x& = & -198 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-198}{-35} \\\Leftrightarrow & x = \frac{198}{35} & & \\ & V = \left\{ \frac{198}{35} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -224x-84 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+224x} & = & \color{red}{-224x} -84 \color{blue}{+224x} \color{blue}{-48} \\\Leftrightarrow & 245x& = & -132 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-132}{245} \\\Leftrightarrow & x = \frac{-132}{245} & & \\ & V = \left\{ \frac{-132}{245} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{2}{5}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{28}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & 28x-420 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{-28x} & = & \color{red}{28x} -420 \color{blue}{-28x} \color{blue}{+20} \\\Leftrightarrow & 7x& = & -400 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-400}{7} \\\Leftrightarrow & x = \frac{-400}{7} & & \\ & V = \left\{ \frac{-400}{7} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{45}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 45x-126 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-45x} & = & \color{red}{45x} -126 \color{blue}{-45x} \color{blue}{-8} \\\Leftrightarrow & -39x& = & -134 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-134}{-39} \\\Leftrightarrow & x = \frac{134}{39} & & \\ & V = \left\{ \frac{134}{39} \right\} & \\\end{align}\)
5. \(\text{40 is het kleinste gemene veelvoud van 2, 8 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{8}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{40.} (\frac{20x}{ \color{blue}{40} }+ \frac{ 15 }{ \color{blue}{40} })& = & (\frac{-56}{ \color{blue}{40} }x-\frac{40}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 20x+15& = & -56x-40 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{+56x} & = & \color{red}{-56x} -40 \color{blue}{+56x} \color{blue}{-15} \\\Leftrightarrow & 76x& = & -55 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{-55}{76} \\\Leftrightarrow & x = \frac{-55}{76} & & \\ & V = \left\{ \frac{-55}{76} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & -72x-630 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{+72x} & = & \color{red}{-72x} -630 \color{blue}{+72x} \color{blue}{-20} \\\Leftrightarrow & 117x& = & -650 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-650}{117} \\\Leftrightarrow & x = \frac{-50}{9} & & \\ & V = \left\{ \frac{-50}{9} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x-420 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} -420 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & -460 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-460}{-21} \\\Leftrightarrow & x = \frac{460}{21} & & \\ & V = \left\{ \frac{460}{21} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -294x+420 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+294x} & = & \color{red}{-294x} +420 \color{blue}{+294x} \color{blue}{+90} \\\Leftrightarrow & 329x& = & 510 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{510}{329} \\\Leftrightarrow & x = \frac{510}{329} & & \\ & V = \left\{ \frac{510}{329} \right\} & \\\end{align}\)
9. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{9}{ \color{blue}{6} }x-\frac{12}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 9x-12 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-9x} & = & \color{red}{9x} -12 \color{blue}{-9x} \color{blue}{+5} \\\Leftrightarrow & -7x& = & -7 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-7}{-7} \\\Leftrightarrow & x = 1 & & \\ & V = \left\{ 1 \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -42x+180 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+42x} & = & \color{red}{-42x} +180 \color{blue}{+42x} \color{blue}{+4} \\\Leftrightarrow & 47x& = & 184 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{184}{47} \\\Leftrightarrow & x = \frac{184}{47} & & \\ & V = \left\{ \frac{184}{47} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & -80x+144 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{+80x} & = & \color{red}{-80x} +144 \color{blue}{+80x} \color{blue}{+15} \\\Leftrightarrow & 104x& = & 159 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{159}{104} \\\Leftrightarrow & x = \frac{159}{104} & & \\ & V = \left\{ \frac{159}{104} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 50 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-50& = & 12x-360 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-12x} & = & \color{red}{12x} -360 \color{blue}{-12x} \color{blue}{+50} \\\Leftrightarrow & 3x& = & -310 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-310}{3} \\\Leftrightarrow & x = \frac{-310}{3} & & \\ & V = \left\{ \frac{-310}{3} \right\} & \\\end{align}\)