Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+16& = & -27x+108 \\\Leftrightarrow & 12x \color{red}{+16} \color{blue}{-16} \color{blue}{+27x} & = & \color{red}{-27x} +108 \color{blue}{+27x} \color{blue}{-16} \\\Leftrightarrow & 39x& = & 92 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{92}{39} \\\Leftrightarrow & x = \frac{92}{39} & & \\ & V = \left\{ \frac{92}{39} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{5}{6}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 35x-42 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-35x} & = & \color{red}{35x} -42 \color{blue}{-35x} \color{blue}{+35} \\\Leftrightarrow & -29x& = & -7 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-7}{-29} \\\Leftrightarrow & x = \frac{7}{29} & & \\ & V = \left\{ \frac{7}{29} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{-110}{ \color{blue}{165} }x+\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-75& = & -110x+330 \\\Leftrightarrow & 33x \color{red}{-75} \color{blue}{+75} \color{blue}{+110x} & = & \color{red}{-110x} +330 \color{blue}{+110x} \color{blue}{+75} \\\Leftrightarrow & 143x& = & 405 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{405}{143} \\\Leftrightarrow & x = \frac{405}{143} & & \\ & V = \left\{ \frac{405}{143} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{1}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 14x-336 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-14x} & = & \color{red}{14x} -336 \color{blue}{-14x} \color{blue}{+12} \\\Leftrightarrow & 7x& = & -324 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-324}{7} \\\Leftrightarrow & x = \frac{-324}{7} & & \\ & V = \left\{ \frac{-324}{7} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 65x+1170 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} +1170 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & 13x& = & 1110 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{1110}{13} \\\Leftrightarrow & x = \frac{1110}{13} & & \\ & V = \left\{ \frac{1110}{13} \right\} & \\\end{align}\)
6. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{56}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 56x+420 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-56x} & = & \color{red}{56x} +420 \color{blue}{-56x} \color{blue}{-80} \\\Leftrightarrow & -21x& = & 340 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{340}{-21} \\\Leftrightarrow & x = \frac{-340}{21} & & \\ & V = \left\{ \frac{-340}{21} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x+84 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & 120 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{120}{7} \\\Leftrightarrow & x = \frac{120}{7} & & \\ & V = \left\{ \frac{120}{7} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -24x-150 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+24x} & = & \color{red}{-24x} -150 \color{blue}{+24x} \color{blue}{-9} \\\Leftrightarrow & 29x& = & -159 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-159}{29} \\\Leftrightarrow & x = \frac{-159}{29} & & \\ & V = \left\{ \frac{-159}{29} \right\} & \\\end{align}\)
9. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x-\frac{630}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & 21x-630 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{-21x} & = & \color{red}{21x} -630 \color{blue}{-21x} \color{blue}{-28} \\\Leftrightarrow & -3x& = & -658 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-658}{-3} \\\Leftrightarrow & x = \frac{658}{3} & & \\ & V = \left\{ \frac{658}{3} \right\} & \\\end{align}\)
10. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{7}{ \color{blue}{21} }x+\frac{147}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & 7x+147 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{-7x} & = & \color{red}{7x} +147 \color{blue}{-7x} \color{blue}{+12} \\\Leftrightarrow & -4x& = & 159 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{159}{-4} \\\Leftrightarrow & x = \frac{-159}{4} & & \\ & V = \left\{ \frac{-159}{4} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 42x-630 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-42x} & = & \color{red}{42x} -630 \color{blue}{-42x} \color{blue}{-120} \\\Leftrightarrow & -7x& = & -750 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-750}{-7} \\\Leftrightarrow & x = \frac{750}{7} & & \\ & V = \left\{ \frac{750}{7} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x-\frac{24}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+15& = & -16x-24 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{+16x} & = & \color{red}{-16x} -24 \color{blue}{+16x} \color{blue}{-15} \\\Leftrightarrow & 22x& = & -39 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{-39}{22} \\\Leftrightarrow & x = \frac{-39}{22} & & \\ & V = \left\{ \frac{-39}{22} \right\} & \\\end{align}\)