Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 147x-210 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-147x} & = & \color{red}{147x} -210 \color{blue}{-147x} \color{blue}{+12} \\\Leftrightarrow & -133x& = & -198 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-198}{-133} \\\Leftrightarrow & x = \frac{198}{133} & & \\ & V = \left\{ \frac{198}{133} \right\} & \\\end{align}\)
2. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 168x-560 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-168x} & = & \color{red}{168x} -560 \color{blue}{-168x} \color{blue}{-80} \\\Leftrightarrow & -133x& = & -640 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-640}{-133} \\\Leftrightarrow & x = \frac{640}{133} & & \\ & V = \left\{ \frac{640}{133} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & 14x-560 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{-14x} & = & \color{red}{14x} -560 \color{blue}{-14x} \color{blue}{+30} \\\Leftrightarrow & 21x& = & -530 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-530}{21} \\\Leftrightarrow & x = \frac{-530}{21} & & \\ & V = \left\{ \frac{-530}{21} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & 88x+132 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{-88x} & = & \color{red}{88x} +132 \color{blue}{-88x} \color{blue}{+24} \\\Leftrightarrow & -55x& = & 156 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{156}{-55} \\\Leftrightarrow & x = \frac{-156}{55} & & \\ & V = \left\{ \frac{-156}{55} \right\} & \\\end{align}\)
5. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{7}{4}x+4 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{245}{ \color{blue}{140} }x+\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-60& = & 245x+560 \\\Leftrightarrow & 28x \color{red}{-60} \color{blue}{+60} \color{blue}{-245x} & = & \color{red}{245x} +560 \color{blue}{-245x} \color{blue}{+60} \\\Leftrightarrow & -217x& = & 620 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{620}{-217} \\\Leftrightarrow & x = \frac{-20}{7} & & \\ & V = \left\{ \frac{-20}{7} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -112x-84 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+112x} & = & \color{red}{-112x} -84 \color{blue}{+112x} \color{blue}{-24} \\\Leftrightarrow & 133x& = & -108 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-108}{133} \\\Leftrightarrow & x = \frac{-108}{133} & & \\ & V = \left\{ \frac{-108}{133} \right\} & \\\end{align}\)
7. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 30 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-30& = & 55x-440 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-55x} & = & \color{red}{55x} -440 \color{blue}{-55x} \color{blue}{+30} \\\Leftrightarrow & -33x& = & -410 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-410}{-33} \\\Leftrightarrow & x = \frac{410}{33} & & \\ & V = \left\{ \frac{410}{33} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-9& = & -70x-210 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{+70x} & = & \color{red}{-70x} -210 \color{blue}{+70x} \color{blue}{+9} \\\Leftrightarrow & 76x& = & -201 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{-201}{76} \\\Leftrightarrow & x = \frac{-201}{76} & & \\ & V = \left\{ \frac{-201}{76} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 52x+624 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-52x} & = & \color{red}{52x} +624 \color{blue}{-52x} \color{blue}{+36} \\\Leftrightarrow & -13x& = & 660 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{660}{-13} \\\Leftrightarrow & x = \frac{-660}{13} & & \\ & V = \left\{ \frac{-660}{13} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & 42x-1050 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{-42x} & = & \color{red}{42x} -1050 \color{blue}{-42x} \color{blue}{+150} \\\Leftrightarrow & -7x& = & -900 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-900}{-7} \\\Leftrightarrow & x = \frac{900}{7} & & \\ & V = \left\{ \frac{900}{7} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-8}{3}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & -128x+384 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{+128x} & = & \color{red}{-128x} +384 \color{blue}{+128x} \color{blue}{-9} \\\Leftrightarrow & 140x& = & 375 \\\Leftrightarrow & \frac{140x}{ \color{red}{140} }& = & \frac{375}{140} \\\Leftrightarrow & x = \frac{75}{28} & & \\ & V = \left\{ \frac{75}{28} \right\} & \\\end{align}\)
12. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-20& = & -27x+108 \\\Leftrightarrow & 12x \color{red}{-20} \color{blue}{+20} \color{blue}{+27x} & = & \color{red}{-27x} +108 \color{blue}{+27x} \color{blue}{+20} \\\Leftrightarrow & 39x& = & 128 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{128}{39} \\\Leftrightarrow & x = \frac{128}{39} & & \\ & V = \left\{ \frac{128}{39} \right\} & \\\end{align}\)