Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{7}{2}x-1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x-\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 49x-14 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-49x} & = & \color{red}{49x} -14 \color{blue}{-49x} \color{blue}{-4} \\\Leftrightarrow & -47x& = & -18 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{-18}{-47} \\\Leftrightarrow & x = \frac{18}{47} & & \\ & V = \left\{ \frac{18}{47} \right\} & \\\end{align}\)
2. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & -112x-160 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{+112x} & = & \color{red}{-112x} -160 \color{blue}{+112x} \color{blue}{-15} \\\Leftrightarrow & 132x& = & -175 \\\Leftrightarrow & \frac{132x}{ \color{red}{132} }& = & \frac{-175}{132} \\\Leftrightarrow & x = \frac{-175}{132} & & \\ & V = \left\{ \frac{-175}{132} \right\} & \\\end{align}\)
3. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{18}{ \color{blue}{45} }x-\frac{135}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-10& = & 18x-135 \\\Leftrightarrow & 15x \color{red}{-10} \color{blue}{+10} \color{blue}{-18x} & = & \color{red}{18x} -135 \color{blue}{-18x} \color{blue}{+10} \\\Leftrightarrow & -3x& = & -125 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-125}{-3} \\\Leftrightarrow & x = \frac{125}{3} & & \\ & V = \left\{ \frac{125}{3} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{3}{2}x-7 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{99}{ \color{blue}{66} }x-\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-18& = & 99x-462 \\\Leftrightarrow & 22x \color{red}{-18} \color{blue}{+18} \color{blue}{-99x} & = & \color{red}{99x} -462 \color{blue}{-99x} \color{blue}{+18} \\\Leftrightarrow & -77x& = & -444 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-444}{-77} \\\Leftrightarrow & x = \frac{444}{77} & & \\ & V = \left\{ \frac{444}{77} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x+490 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} +490 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & 470 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{470}{-21} \\\Leftrightarrow & x = \frac{-470}{21} & & \\ & V = \left\{ \frac{-470}{21} \right\} & \\\end{align}\)
6. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{-3}{4}x+5 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-135}{ \color{blue}{180} }x+\frac{900}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x+40& = & -135x+900 \\\Leftrightarrow & 36x \color{red}{+40} \color{blue}{-40} \color{blue}{+135x} & = & \color{red}{-135x} +900 \color{blue}{+135x} \color{blue}{-40} \\\Leftrightarrow & 171x& = & 860 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{860}{171} \\\Leftrightarrow & x = \frac{860}{171} & & \\ & V = \left\{ \frac{860}{171} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{7}{6}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{49}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+18& = & 49x+294 \\\Leftrightarrow & 6x \color{red}{+18} \color{blue}{-18} \color{blue}{-49x} & = & \color{red}{49x} +294 \color{blue}{-49x} \color{blue}{-18} \\\Leftrightarrow & -43x& = & 276 \\\Leftrightarrow & \frac{-43x}{ \color{red}{-43} }& = & \frac{276}{-43} \\\Leftrightarrow & x = \frac{-276}{43} & & \\ & V = \left\{ \frac{-276}{43} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{3}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 63x-420 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-63x} & = & \color{red}{63x} -420 \color{blue}{-63x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & -396 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-396}{-35} \\\Leftrightarrow & x = \frac{396}{35} & & \\ & V = \left\{ \frac{396}{35} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x-\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 6x-54 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} -54 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & 3x& = & -58 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-58}{3} \\\Leftrightarrow & x = \frac{-58}{3} & & \\ & V = \left\{ \frac{-58}{3} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 252x+1470 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-252x} & = & \color{red}{252x} +1470 \color{blue}{-252x} \color{blue}{-60} \\\Leftrightarrow & -217x& = & 1410 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{1410}{-217} \\\Leftrightarrow & x = \frac{-1410}{217} & & \\ & V = \left\{ \frac{-1410}{217} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 105x-168 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-105x} & = & \color{red}{105x} -168 \color{blue}{-105x} \color{blue}{-18} \\\Leftrightarrow & -91x& = & -186 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-186}{-91} \\\Leftrightarrow & x = \frac{186}{91} & & \\ & V = \left\{ \frac{186}{91} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x+\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+80& = & -245x+560 \\\Leftrightarrow & 28x \color{red}{+80} \color{blue}{-80} \color{blue}{+245x} & = & \color{red}{-245x} +560 \color{blue}{+245x} \color{blue}{-80} \\\Leftrightarrow & 273x& = & 480 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{480}{273} \\\Leftrightarrow & x = \frac{160}{91} & & \\ & V = \left\{ \frac{160}{91} \right\} & \\\end{align}\)