Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}-\frac{3}{7}=\frac{7}{3}x-2\)
2. \(\frac{x}{4}-\frac{5}{8}=\frac{5}{3}x-8\)
3. \(\frac{x}{5}+\frac{2}{13}=\frac{-3}{4}x+3\)
4. \(\frac{x}{4}-\frac{3}{16}=\frac{-8}{3}x-7\)
5. \(\frac{x}{6}+\frac{2}{7}=\frac{6}{5}x+5\)
6. \(\frac{x}{2}-\frac{5}{16}=\frac{4}{3}x+5\)
7. \(\frac{x}{4}-\frac{5}{14}=\frac{7}{3}x-5\)
8. \(\frac{x}{5}+\frac{4}{11}=\frac{2}{3}x-2\)
9. \(\frac{x}{6}-\frac{4}{7}=\frac{4}{5}x+5\)
10. \(\frac{x}{2}+\frac{2}{7}=\frac{4}{3}x+2\)
11. \(\frac{x}{7}+\frac{2}{7}=\frac{-2}{5}x-2\)
12. \(\frac{x}{2}+\frac{3}{14}=\frac{1}{5}x-6\)

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{7}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 98x-84 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-98x} & = & \color{red}{98x} -84 \color{blue}{-98x} \color{blue}{+18} \\\Leftrightarrow & -77x& = & -66 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-66}{-77} \\\Leftrightarrow & x = \frac{6}{7} & & \\ & V = \left\{ \frac{6}{7} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{5}{3}x-8 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{40}{ \color{blue}{24} }x-\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-15& = & 40x-192 \\\Leftrightarrow & 6x \color{red}{-15} \color{blue}{+15} \color{blue}{-40x} & = & \color{red}{40x} -192 \color{blue}{-40x} \color{blue}{+15} \\\Leftrightarrow & -34x& = & -177 \\\Leftrightarrow & \frac{-34x}{ \color{red}{-34} }& = & \frac{-177}{-34} \\\Leftrightarrow & x = \frac{177}{34} & & \\ & V = \left\{ \frac{177}{34} \right\} & \\\end{align}\)
3. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{-195}{ \color{blue}{260} }x+\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+40& = & -195x+780 \\\Leftrightarrow & 52x \color{red}{+40} \color{blue}{-40} \color{blue}{+195x} & = & \color{red}{-195x} +780 \color{blue}{+195x} \color{blue}{-40} \\\Leftrightarrow & 247x& = & 740 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{740}{247} \\\Leftrightarrow & x = \frac{740}{247} & & \\ & V = \left\{ \frac{740}{247} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{-8}{3}x-7 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x-\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-9& = & -128x-336 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{+128x} & = & \color{red}{-128x} -336 \color{blue}{+128x} \color{blue}{+9} \\\Leftrightarrow & 140x& = & -327 \\\Leftrightarrow & \frac{140x}{ \color{red}{140} }& = & \frac{-327}{140} \\\Leftrightarrow & x = \frac{-327}{140} & & \\ & V = \left\{ \frac{-327}{140} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 252x+1050 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-252x} & = & \color{red}{252x} +1050 \color{blue}{-252x} \color{blue}{-60} \\\Leftrightarrow & -217x& = & 990 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{990}{-217} \\\Leftrightarrow & x = \frac{-990}{217} & & \\ & V = \left\{ \frac{-990}{217} \right\} & \\\end{align}\)
6. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & 64x+240 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{-64x} & = & \color{red}{64x} +240 \color{blue}{-64x} \color{blue}{+15} \\\Leftrightarrow & -40x& = & 255 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{255}{-40} \\\Leftrightarrow & x = \frac{-51}{8} & & \\ & V = \left\{ \frac{-51}{8} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{14}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 30 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-30& = & 196x-420 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-196x} & = & \color{red}{196x} -420 \color{blue}{-196x} \color{blue}{+30} \\\Leftrightarrow & -175x& = & -390 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-390}{-175} \\\Leftrightarrow & x = \frac{78}{35} & & \\ & V = \left\{ \frac{78}{35} \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{2}{3}x-2 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 60 }{ \color{blue}{165} })& = & (\frac{110}{ \color{blue}{165} }x-\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+60& = & 110x-330 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-110x} & = & \color{red}{110x} -330 \color{blue}{-110x} \color{blue}{-60} \\\Leftrightarrow & -77x& = & -390 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-390}{-77} \\\Leftrightarrow & x = \frac{390}{77} & & \\ & V = \left\{ \frac{390}{77} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{4}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 168x+1050 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-168x} & = & \color{red}{168x} +1050 \color{blue}{-168x} \color{blue}{+120} \\\Leftrightarrow & -133x& = & 1170 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{1170}{-133} \\\Leftrightarrow & x = \frac{-1170}{133} & & \\ & V = \left\{ \frac{-1170}{133} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 56x+84 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-56x} & = & \color{red}{56x} +84 \color{blue}{-56x} \color{blue}{-12} \\\Leftrightarrow & -35x& = & 72 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{72}{-35} \\\Leftrightarrow & x = \frac{-72}{35} & & \\ & V = \left\{ \frac{-72}{35} \right\} & \\\end{align}\)
11. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 10 }{ \color{blue}{35} })& = & (\frac{-14}{ \color{blue}{35} }x-\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+10& = & -14x-70 \\\Leftrightarrow & 5x \color{red}{+10} \color{blue}{-10} \color{blue}{+14x} & = & \color{red}{-14x} -70 \color{blue}{+14x} \color{blue}{-10} \\\Leftrightarrow & 19x& = & -80 \\\Leftrightarrow & \frac{19x}{ \color{red}{19} }& = & \frac{-80}{19} \\\Leftrightarrow & x = \frac{-80}{19} & & \\ & V = \left\{ \frac{-80}{19} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{14}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 15 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+15& = & 14x-420 \\\Leftrightarrow & 35x \color{red}{+15} \color{blue}{-15} \color{blue}{-14x} & = & \color{red}{14x} -420 \color{blue}{-14x} \color{blue}{-15} \\\Leftrightarrow & 21x& = & -435 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-435}{21} \\\Leftrightarrow & x = \frac{-145}{7} & & \\ & V = \left\{ \frac{-145}{7} \right\} & \\\end{align}\)