Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{40.} (\frac{10x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{48}{ \color{blue}{40} }x-\frac{200}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 10x-25& = & 48x-200 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-48x} & = & \color{red}{48x} -200 \color{blue}{-48x} \color{blue}{+25} \\\Leftrightarrow & -38x& = & -175 \\\Leftrightarrow & \frac{-38x}{ \color{red}{-38} }& = & \frac{-175}{-38} \\\Leftrightarrow & x = \frac{175}{38} & & \\ & V = \left\{ \frac{175}{38} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & -168x-1680 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{+168x} & = & \color{red}{-168x} -1680 \color{blue}{+168x} \color{blue}{+120} \\\Leftrightarrow & 203x& = & -1560 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1560}{203} \\\Leftrightarrow & x = \frac{-1560}{203} & & \\ & V = \left\{ \frac{-1560}{203} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 60 }{ \color{blue}{165} })& = & (\frac{99}{ \color{blue}{165} }x-\frac{990}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-60& = & 99x-990 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-99x} & = & \color{red}{99x} -990 \color{blue}{-99x} \color{blue}{+60} \\\Leftrightarrow & -44x& = & -930 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-930}{-44} \\\Leftrightarrow & x = \frac{465}{22} & & \\ & V = \left\{ \frac{465}{22} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 112x+336 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-112x} & = & \color{red}{112x} +336 \color{blue}{-112x} \color{blue}{-9} \\\Leftrightarrow & -88x& = & 327 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{327}{-88} \\\Leftrightarrow & x = \frac{-327}{88} & & \\ & V = \left\{ \frac{-327}{88} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{3}{2}x-1 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{165}{ \color{blue}{110} }x-\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-20& = & 165x-110 \\\Leftrightarrow & 22x \color{red}{-20} \color{blue}{+20} \color{blue}{-165x} & = & \color{red}{165x} -110 \color{blue}{-165x} \color{blue}{+20} \\\Leftrightarrow & -143x& = & -90 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-90}{-143} \\\Leftrightarrow & x = \frac{90}{143} & & \\ & V = \left\{ \frac{90}{143} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-20}{ \color{blue}{12} }x+\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+10& = & -20x+36 \\\Leftrightarrow & 3x \color{red}{+10} \color{blue}{-10} \color{blue}{+20x} & = & \color{red}{-20x} +36 \color{blue}{+20x} \color{blue}{-10} \\\Leftrightarrow & 23x& = & 26 \\\Leftrightarrow & \frac{23x}{ \color{red}{23} }& = & \frac{26}{23} \\\Leftrightarrow & x = \frac{26}{23} & & \\ & V = \left\{ \frac{26}{23} \right\} & \\\end{align}\)
7. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x+\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+60& = & -70x+525 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{+70x} & = & \color{red}{-70x} +525 \color{blue}{+70x} \color{blue}{-60} \\\Leftrightarrow & 91x& = & 465 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{465}{91} \\\Leftrightarrow & x = \frac{465}{91} & & \\ & V = \left\{ \frac{465}{91} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 208x+156 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-208x} & = & \color{red}{208x} +156 \color{blue}{-208x} \color{blue}{+36} \\\Leftrightarrow & -169x& = & 192 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{192}{-169} \\\Leftrightarrow & x = \frac{-192}{169} & & \\ & V = \left\{ \frac{-192}{169} \right\} & \\\end{align}\)
9. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{-49}{ \color{blue}{35} }x+\frac{175}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & -49x+175 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{+49x} & = & \color{red}{-49x} +175 \color{blue}{+49x} \color{blue}{-20} \\\Leftrightarrow & 54x& = & 155 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{155}{54} \\\Leftrightarrow & x = \frac{155}{54} & & \\ & V = \left\{ \frac{155}{54} \right\} & \\\end{align}\)
10. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x+90 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} +90 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & 86 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{86}{-3} \\\Leftrightarrow & x = \frac{-86}{3} & & \\ & V = \left\{ \frac{-86}{3} \right\} & \\\end{align}\)
11. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x-\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+16& = & -27x-180 \\\Leftrightarrow & 12x \color{red}{+16} \color{blue}{-16} \color{blue}{+27x} & = & \color{red}{-27x} -180 \color{blue}{+27x} \color{blue}{-16} \\\Leftrightarrow & 39x& = & -196 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-196}{39} \\\Leftrightarrow & x = \frac{-196}{39} & & \\ & V = \left\{ \frac{-196}{39} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x+\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & -112x+140 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+112x} & = & \color{red}{-112x} +140 \color{blue}{+112x} \color{blue}{-40} \\\Leftrightarrow & 147x& = & 100 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{100}{147} \\\Leftrightarrow & x = \frac{100}{147} & & \\ & V = \left\{ \frac{100}{147} \right\} & \\\end{align}\)