Vgln. eerste graad (reeks 4)

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

1. \(\frac{x}{7}+\frac{4}{15}=\frac{-5}{3}x-7\)
2. \(\frac{x}{2}+\frac{5}{9}=\frac{1}{3}x-4\)
3. \(\frac{x}{5}-\frac{4}{7}=\frac{-3}{4}x+1\)
4. \(\frac{x}{4}+\frac{3}{13}=\frac{5}{3}x-1\)
5. \(\frac{x}{2}+\frac{4}{11}=\frac{1}{5}x+7\)
6. \(\frac{x}{3}+\frac{3}{16}=\frac{1}{2}x+8\)
7. \(\frac{x}{6}+\frac{4}{13}=\frac{-2}{5}x+1\)
8. \(\frac{x}{4}-\frac{4}{7}=\frac{7}{3}x-1\)
9. \(\frac{x}{3}+\frac{3}{10}=\frac{7}{4}x-7\)
10. \(\frac{x}{7}-\frac{3}{10}=\frac{-3}{4}x-5\)
11. \(\frac{x}{4}+\frac{5}{14}=\frac{-2}{3}x+2\)
12. \(\frac{x}{7}+\frac{2}{7}=\frac{3}{4}x+6\)

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

Verbetersleutel

1. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & -175x-735 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{+175x} & = & \color{red}{-175x} -735 \color{blue}{+175x} \color{blue}{-28} \\\Leftrightarrow & 190x& = & -763 \\\Leftrightarrow & \frac{190x}{ \color{red}{190} }& = & \frac{-763}{190} \\\Leftrightarrow & x = \frac{-763}{190} & & \\ & V = \left\{ \frac{-763}{190} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x-\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & 6x-72 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{-6x} & = & \color{red}{6x} -72 \color{blue}{-6x} \color{blue}{-10} \\\Leftrightarrow & 3x& = & -82 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-82}{3} \\\Leftrightarrow & x = \frac{-82}{3} & & \\ & V = \left\{ \frac{-82}{3} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x+\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-80& = & -105x+140 \\\Leftrightarrow & 28x \color{red}{-80} \color{blue}{+80} \color{blue}{+105x} & = & \color{red}{-105x} +140 \color{blue}{+105x} \color{blue}{+80} \\\Leftrightarrow & 133x& = & 220 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{220}{133} \\\Leftrightarrow & x = \frac{220}{133} & & \\ & V = \left\{ \frac{220}{133} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{260}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & 260x-156 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{-260x} & = & \color{red}{260x} -156 \color{blue}{-260x} \color{blue}{-36} \\\Leftrightarrow & -221x& = & -192 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{-192}{-221} \\\Leftrightarrow & x = \frac{192}{221} & & \\ & V = \left\{ \frac{192}{221} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x+\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+40& = & 22x+770 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{-22x} & = & \color{red}{22x} +770 \color{blue}{-22x} \color{blue}{-40} \\\Leftrightarrow & 33x& = & 730 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{730}{33} \\\Leftrightarrow & x = \frac{730}{33} & & \\ & V = \left\{ \frac{730}{33} \right\} & \\\end{align}\)
6. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x+384 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} +384 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & 375 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{375}{-8} \\\Leftrightarrow & x = \frac{-375}{8} & & \\ & V = \left\{ \frac{-375}{8} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -156x+390 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+156x} & = & \color{red}{-156x} +390 \color{blue}{+156x} \color{blue}{-120} \\\Leftrightarrow & 221x& = & 270 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{270}{221} \\\Leftrightarrow & x = \frac{270}{221} & & \\ & V = \left\{ \frac{270}{221} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 196x-84 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-196x} & = & \color{red}{196x} -84 \color{blue}{-196x} \color{blue}{+48} \\\Leftrightarrow & -175x& = & -36 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-36}{-175} \\\Leftrightarrow & x = \frac{36}{175} & & \\ & V = \left\{ \frac{36}{175} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{7}{4}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+18& = & 105x-420 \\\Leftrightarrow & 20x \color{red}{+18} \color{blue}{-18} \color{blue}{-105x} & = & \color{red}{105x} -420 \color{blue}{-105x} \color{blue}{-18} \\\Leftrightarrow & -85x& = & -438 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-438}{-85} \\\Leftrightarrow & x = \frac{438}{85} & & \\ & V = \left\{ \frac{438}{85} \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 7, 10 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{140.} (\frac{20x}{ \color{blue}{140} }- \frac{ 42 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 20x-42& = & -105x-700 \\\Leftrightarrow & 20x \color{red}{-42} \color{blue}{+42} \color{blue}{+105x} & = & \color{red}{-105x} -700 \color{blue}{+105x} \color{blue}{+42} \\\Leftrightarrow & 125x& = & -658 \\\Leftrightarrow & \frac{125x}{ \color{red}{125} }& = & \frac{-658}{125} \\\Leftrightarrow & x = \frac{-658}{125} & & \\ & V = \left\{ \frac{-658}{125} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{14}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 30 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+30& = & -56x+168 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{+56x} & = & \color{red}{-56x} +168 \color{blue}{+56x} \color{blue}{-30} \\\Leftrightarrow & 77x& = & 138 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{138}{77} \\\Leftrightarrow & x = \frac{138}{77} & & \\ & V = \left\{ \frac{138}{77} \right\} & \\\end{align}\)
12. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{3}{4}x+6 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 8 }{ \color{blue}{28} })& = & (\frac{21}{ \color{blue}{28} }x+\frac{168}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+8& = & 21x+168 \\\Leftrightarrow & 4x \color{red}{+8} \color{blue}{-8} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{-8} \\\Leftrightarrow & -17x& = & 160 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{160}{-17} \\\Leftrightarrow & x = \frac{-160}{17} & & \\ & V = \left\{ \frac{-160}{17} \right\} & \\\end{align}\)