Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{4}{5}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{48}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 48x+480 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-48x} & = & \color{red}{48x} +480 \color{blue}{-48x} \color{blue}{+16} \\\Leftrightarrow & -33x& = & 496 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{496}{-33} \\\Leftrightarrow & x = \frac{-496}{33} & & \\ & V = \left\{ \frac{-496}{33} \right\} & \\\end{align}\)
2. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }- \frac{ 4 }{ \color{blue}{15} })& = & (\frac{3}{ \color{blue}{15} }x-\frac{45}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x-4& = & 3x-45 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-3x} & = & \color{red}{3x} -45 \color{blue}{-3x} \color{blue}{+4} \\\Leftrightarrow & 2x& = & -41 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{-41}{2} \\\Leftrightarrow & x = \frac{-41}{2} & & \\ & V = \left\{ \frac{-41}{2} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-20}{ \color{blue}{12} }x-\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+5& = & -20x-60 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{+20x} & = & \color{red}{-20x} -60 \color{blue}{+20x} \color{blue}{-5} \\\Leftrightarrow & 23x& = & -65 \\\Leftrightarrow & \frac{23x}{ \color{red}{23} }& = & \frac{-65}{23} \\\Leftrightarrow & x = \frac{-65}{23} & & \\ & V = \left\{ \frac{-65}{23} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 28x+420 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-28x} & = & \color{red}{28x} +420 \color{blue}{-28x} \color{blue}{+48} \\\Leftrightarrow & -7x& = & 468 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{468}{-7} \\\Leftrightarrow & x = \frac{-468}{7} & & \\ & V = \left\{ \frac{-468}{7} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{12}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 12x+144 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} +144 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & 4x& = & 153 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{153}{4} \\\Leftrightarrow & x = \frac{153}{4} & & \\ & V = \left\{ \frac{153}{4} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x+\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+24& = & -104x+1092 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+104x} & = & \color{red}{-104x} +1092 \color{blue}{+104x} \color{blue}{-24} \\\Leftrightarrow & 143x& = & 1068 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{1068}{143} \\\Leftrightarrow & x = \frac{1068}{143} & & \\ & V = \left\{ \frac{1068}{143} \right\} & \\\end{align}\)
7. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{3}{2}x+4 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{36}{ \color{blue}{24} }x+\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 36x+96 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-36x} & = & \color{red}{36x} +96 \color{blue}{-36x} \color{blue}{+9} \\\Leftrightarrow & -28x& = & 105 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{105}{-28} \\\Leftrightarrow & x = \frac{-15}{4} & & \\ & V = \left\{ \frac{-15}{4} \right\} & \\\end{align}\)
8. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{315.} (\frac{45x}{ \color{blue}{315} }+ \frac{ 70 }{ \color{blue}{315} })& = & (\frac{63}{ \color{blue}{315} }x+\frac{1575}{ \color{blue}{315} }) \color{blue}{.315} \\\Leftrightarrow & 45x+70& = & 63x+1575 \\\Leftrightarrow & 45x \color{red}{+70} \color{blue}{-70} \color{blue}{-63x} & = & \color{red}{63x} +1575 \color{blue}{-63x} \color{blue}{-70} \\\Leftrightarrow & -18x& = & 1505 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{1505}{-18} \\\Leftrightarrow & x = \frac{-1505}{18} & & \\ & V = \left\{ \frac{-1505}{18} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{7}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+40& = & 126x-90 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{-126x} & = & \color{red}{126x} -90 \color{blue}{-126x} \color{blue}{-40} \\\Leftrightarrow & -81x& = & -130 \\\Leftrightarrow & \frac{-81x}{ \color{red}{-81} }& = & \frac{-130}{-81} \\\Leftrightarrow & x = \frac{130}{81} & & \\ & V = \left\{ \frac{130}{81} \right\} & \\\end{align}\)
10. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{-56}{ \color{blue}{21} }x+\frac{21}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & -56x+21 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{+56x} & = & \color{red}{-56x} +21 \color{blue}{+56x} \color{blue}{-9} \\\Leftrightarrow & 59x& = & 12 \\\Leftrightarrow & \frac{59x}{ \color{red}{59} }& = & \frac{12}{59} \\\Leftrightarrow & x = \frac{12}{59} & & \\ & V = \left\{ \frac{12}{59} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & -63x-252 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{+63x} & = & \color{red}{-63x} -252 \color{blue}{+63x} \color{blue}{+36} \\\Leftrightarrow & 91x& = & -216 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-216}{91} \\\Leftrightarrow & x = \frac{-216}{91} & & \\ & V = \left\{ \frac{-216}{91} \right\} & \\\end{align}\)
12. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{9}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 20 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x+\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+20& = & -60x+72 \\\Leftrightarrow & 9x \color{red}{+20} \color{blue}{-20} \color{blue}{+60x} & = & \color{red}{-60x} +72 \color{blue}{+60x} \color{blue}{-20} \\\Leftrightarrow & 69x& = & 52 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{52}{69} \\\Leftrightarrow & x = \frac{52}{69} & & \\ & V = \left\{ \frac{52}{69} \right\} & \\\end{align}\)