Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{4}x-1 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 12 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x-\frac{28}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+12& = & 7x-28 \\\Leftrightarrow & 4x \color{red}{+12} \color{blue}{-12} \color{blue}{-7x} & = & \color{red}{7x} -28 \color{blue}{-7x} \color{blue}{-12} \\\Leftrightarrow & -3x& = & -40 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-40}{-3} \\\Leftrightarrow & x = \frac{40}{3} & & \\ & V = \left\{ \frac{40}{3} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{8}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-40}{ \color{blue}{24} }x-\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-9& = & -40x-72 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{+40x} & = & \color{red}{-40x} -72 \color{blue}{+40x} \color{blue}{+9} \\\Leftrightarrow & 46x& = & -63 \\\Leftrightarrow & \frac{46x}{ \color{red}{46} }& = & \frac{-63}{46} \\\Leftrightarrow & x = \frac{-63}{46} & & \\ & V = \left\{ \frac{-63}{46} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 3, 14 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{70x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 70x+45& = & 42x+1680 \\\Leftrightarrow & 70x \color{red}{+45} \color{blue}{-45} \color{blue}{-42x} & = & \color{red}{42x} +1680 \color{blue}{-42x} \color{blue}{-45} \\\Leftrightarrow & 28x& = & 1635 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{1635}{28} \\\Leftrightarrow & x = \frac{1635}{28} & & \\ & V = \left\{ \frac{1635}{28} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }- \frac{ 80 }{ \color{blue}{220} })& = & (\frac{-385}{ \color{blue}{220} }x+\frac{1540}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x-80& = & -385x+1540 \\\Leftrightarrow & 44x \color{red}{-80} \color{blue}{+80} \color{blue}{+385x} & = & \color{red}{-385x} +1540 \color{blue}{+385x} \color{blue}{+80} \\\Leftrightarrow & 429x& = & 1620 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{1620}{429} \\\Leftrightarrow & x = \frac{540}{143} & & \\ & V = \left\{ \frac{540}{143} \right\} & \\\end{align}\)
5. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }- \frac{ 56 }{ \color{blue}{308} })& = & (\frac{77}{ \color{blue}{308} }x+\frac{924}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x-56& = & 77x+924 \\\Leftrightarrow & 44x \color{red}{-56} \color{blue}{+56} \color{blue}{-77x} & = & \color{red}{77x} +924 \color{blue}{-77x} \color{blue}{+56} \\\Leftrightarrow & -33x& = & 980 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{980}{-33} \\\Leftrightarrow & x = \frac{-980}{33} & & \\ & V = \left\{ \frac{-980}{33} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+36& = & -63x-504 \\\Leftrightarrow & 28x \color{red}{+36} \color{blue}{-36} \color{blue}{+63x} & = & \color{red}{-63x} -504 \color{blue}{+63x} \color{blue}{-36} \\\Leftrightarrow & 91x& = & -540 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-540}{91} \\\Leftrightarrow & x = \frac{-540}{91} & & \\ & V = \left\{ \frac{-540}{91} \right\} & \\\end{align}\)
7. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{-7}{4}x-2 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 20 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x-\frac{56}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-20& = & -49x-56 \\\Leftrightarrow & 4x \color{red}{-20} \color{blue}{+20} \color{blue}{+49x} & = & \color{red}{-49x} -56 \color{blue}{+49x} \color{blue}{+20} \\\Leftrightarrow & 53x& = & -36 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{-36}{53} \\\Leftrightarrow & x = \frac{-36}{53} & & \\ & V = \left\{ \frac{-36}{53} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & -112x-280 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+112x} & = & \color{red}{-112x} -280 \color{blue}{+112x} \color{blue}{-60} \\\Leftrightarrow & 147x& = & -340 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-340}{147} \\\Leftrightarrow & x = \frac{-340}{147} & & \\ & V = \left\{ \frac{-340}{147} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & -40x-60 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{+40x} & = & \color{red}{-40x} -60 \color{blue}{+40x} \color{blue}{+25} \\\Leftrightarrow & 52x& = & -35 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{-35}{52} \\\Leftrightarrow & x = \frac{-35}{52} & & \\ & V = \left\{ \frac{-35}{52} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{36}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & 36x+480 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-36x} & = & \color{red}{36x} +480 \color{blue}{-36x} \color{blue}{-25} \\\Leftrightarrow & -26x& = & 455 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{455}{-26} \\\Leftrightarrow & x = \frac{-35}{2} & & \\ & V = \left\{ \frac{-35}{2} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x+336 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} +336 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 360 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{360}{-7} \\\Leftrightarrow & x = \frac{-360}{7} & & \\ & V = \left\{ \frac{-360}{7} \right\} & \\\end{align}\)
12. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 9x-126 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-9x} & = & \color{red}{9x} -126 \color{blue}{-9x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & -134 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-134}{-3} \\\Leftrightarrow & x = \frac{134}{3} & & \\ & V = \left\{ \frac{134}{3} \right\} & \\\end{align}\)