Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}-\frac{3}{14}=\frac{-7}{4}x+2\)
2. \(\frac{x}{3}+\frac{2}{7}=\frac{1}{4}x+2\)
3. \(\frac{x}{6}-\frac{4}{15}=\frac{-2}{5}x+3\)
4. \(\frac{x}{3}+\frac{2}{13}=\frac{1}{4}x-6\)
5. \(\frac{x}{4}-\frac{5}{11}=\frac{-5}{3}x+6\)
6. \(\frac{x}{4}-\frac{2}{15}=\frac{6}{5}x+8\)
7. \(\frac{x}{5}+\frac{2}{15}=\frac{1}{6}x+5\)
8. \(\frac{x}{5}+\frac{2}{11}=\frac{-3}{4}x-7\)
9. \(\frac{x}{5}+\frac{5}{14}=\frac{-5}{6}x-1\)
10. \(\frac{x}{6}+\frac{3}{8}=\frac{6}{5}x+4\)
11. \(\frac{x}{4}+\frac{4}{7}=\frac{-5}{3}x+3\)
12. \(\frac{x}{6}+\frac{3}{7}=\frac{-2}{5}x-3\)

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 5, 14 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{-7}{4}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 30 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-30& = & -245x+280 \\\Leftrightarrow & 28x \color{red}{-30} \color{blue}{+30} \color{blue}{+245x} & = & \color{red}{-245x} +280 \color{blue}{+245x} \color{blue}{+30} \\\Leftrightarrow & 273x& = & 310 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{310}{273} \\\Leftrightarrow & x = \frac{310}{273} & & \\ & V = \left\{ \frac{310}{273} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+24& = & 21x+168 \\\Leftrightarrow & 28x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & 7x& = & 144 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{144}{7} \\\Leftrightarrow & x = \frac{144}{7} & & \\ & V = \left\{ \frac{144}{7} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -12x+90 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+12x} & = & \color{red}{-12x} +90 \color{blue}{+12x} \color{blue}{+8} \\\Leftrightarrow & 17x& = & 98 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{98}{17} \\\Leftrightarrow & x = \frac{98}{17} & & \\ & V = \left\{ \frac{98}{17} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x-\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+24& = & 39x-936 \\\Leftrightarrow & 52x \color{red}{+24} \color{blue}{-24} \color{blue}{-39x} & = & \color{red}{39x} -936 \color{blue}{-39x} \color{blue}{-24} \\\Leftrightarrow & 13x& = & -960 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-960}{13} \\\Leftrightarrow & x = \frac{-960}{13} & & \\ & V = \left\{ \frac{-960}{13} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-60& = & -220x+792 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{+220x} & = & \color{red}{-220x} +792 \color{blue}{+220x} \color{blue}{+60} \\\Leftrightarrow & 253x& = & 852 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{852}{253} \\\Leftrightarrow & x = \frac{852}{253} & & \\ & V = \left\{ \frac{852}{253} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 72x+480 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-72x} & = & \color{red}{72x} +480 \color{blue}{-72x} \color{blue}{+8} \\\Leftrightarrow & -57x& = & 488 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{488}{-57} \\\Leftrightarrow & x = \frac{-488}{57} & & \\ & V = \left\{ \frac{-488}{57} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 5x+150 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-5x} & = & \color{red}{5x} +150 \color{blue}{-5x} \color{blue}{-4} \\\Leftrightarrow & x& = & 146 \\ & V = \left\{ 146 \right\} & \\\end{align}\)
8. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-165}{ \color{blue}{220} }x-\frac{1540}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+40& = & -165x-1540 \\\Leftrightarrow & 44x \color{red}{+40} \color{blue}{-40} \color{blue}{+165x} & = & \color{red}{-165x} -1540 \color{blue}{+165x} \color{blue}{-40} \\\Leftrightarrow & 209x& = & -1580 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-1580}{209} \\\Leftrightarrow & x = \frac{-1580}{209} & & \\ & V = \left\{ \frac{-1580}{209} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{-5}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+75& = & -175x-210 \\\Leftrightarrow & 42x \color{red}{+75} \color{blue}{-75} \color{blue}{+175x} & = & \color{red}{-175x} -210 \color{blue}{+175x} \color{blue}{-75} \\\Leftrightarrow & 217x& = & -285 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-285}{217} \\\Leftrightarrow & x = \frac{-285}{217} & & \\ & V = \left\{ \frac{-285}{217} \right\} & \\\end{align}\)
10. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{144}{ \color{blue}{120} }x+\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & 144x+480 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{-144x} & = & \color{red}{144x} +480 \color{blue}{-144x} \color{blue}{-45} \\\Leftrightarrow & -124x& = & 435 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{435}{-124} \\\Leftrightarrow & x = \frac{-435}{124} & & \\ & V = \left\{ \frac{-435}{124} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -140x+252 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+140x} & = & \color{red}{-140x} +252 \color{blue}{+140x} \color{blue}{-48} \\\Leftrightarrow & 161x& = & 204 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{204}{161} \\\Leftrightarrow & x = \frac{204}{161} & & \\ & V = \left\{ \frac{204}{161} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -84x-630 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+84x} & = & \color{red}{-84x} -630 \color{blue}{+84x} \color{blue}{-90} \\\Leftrightarrow & 119x& = & -720 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-720}{119} \\\Leftrightarrow & x = \frac{-720}{119} & & \\ & V = \left\{ \frac{-720}{119} \right\} & \\\end{align}\)