Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{6}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-24& = & 7x+294 \\\Leftrightarrow & 6x \color{red}{-24} \color{blue}{+24} \color{blue}{-7x} & = & \color{red}{7x} +294 \color{blue}{-7x} \color{blue}{+24} \\\Leftrightarrow & -x& = & 318 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{318}{-1} \\\Leftrightarrow & x = -318 & & \\ & V = \left\{ -318 \right\} & \\\end{align}\)
2. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }+ \frac{ 20 }{ \color{blue}{45} })& = & (\frac{15}{ \color{blue}{45} }x-\frac{90}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x+20& = & 15x-90 \\\Leftrightarrow & 9x \color{red}{+20} \color{blue}{-20} \color{blue}{-15x} & = & \color{red}{15x} -90 \color{blue}{-15x} \color{blue}{-20} \\\Leftrightarrow & -6x& = & -110 \\\Leftrightarrow & \frac{-6x}{ \color{red}{-6} }& = & \frac{-110}{-6} \\\Leftrightarrow & x = \frac{55}{3} & & \\ & V = \left\{ \frac{55}{3} \right\} & \\\end{align}\)
3. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x-\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & -168x-600 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{+168x} & = & \color{red}{-168x} -600 \color{blue}{+168x} \color{blue}{+75} \\\Leftrightarrow & 188x& = & -525 \\\Leftrightarrow & \frac{188x}{ \color{red}{188} }& = & \frac{-525}{188} \\\Leftrightarrow & x = \frac{-525}{188} & & \\ & V = \left\{ \frac{-525}{188} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & -80x+384 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{+80x} & = & \color{red}{-80x} +384 \color{blue}{+80x} \color{blue}{-9} \\\Leftrightarrow & 92x& = & 375 \\\Leftrightarrow & \frac{92x}{ \color{red}{92} }& = & \frac{375}{92} \\\Leftrightarrow & x = \frac{375}{92} & & \\ & V = \left\{ \frac{375}{92} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 112x-588 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-112x} & = & \color{red}{112x} -588 \color{blue}{-112x} \color{blue}{-36} \\\Leftrightarrow & -91x& = & -624 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-624}{-91} \\\Leftrightarrow & x = \frac{48}{7} & & \\ & V = \left\{ \frac{48}{7} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{11}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 30 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-30& = & 55x+660 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-55x} & = & \color{red}{55x} +660 \color{blue}{-55x} \color{blue}{+30} \\\Leftrightarrow & -33x& = & 690 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{690}{-33} \\\Leftrightarrow & x = \frac{-230}{11} & & \\ & V = \left\{ \frac{-230}{11} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x-\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+15& = & -128x-48 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+128x} & = & \color{red}{-128x} -48 \color{blue}{+128x} \color{blue}{-15} \\\Leftrightarrow & 140x& = & -63 \\\Leftrightarrow & \frac{140x}{ \color{red}{140} }& = & \frac{-63}{140} \\\Leftrightarrow & x = \frac{-9}{20} & & \\ & V = \left\{ \frac{-9}{20} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & 72x+120 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-72x} & = & \color{red}{72x} +120 \color{blue}{-72x} \color{blue}{+25} \\\Leftrightarrow & -62x& = & 145 \\\Leftrightarrow & \frac{-62x}{ \color{red}{-62} }& = & \frac{145}{-62} \\\Leftrightarrow & x = \frac{-145}{62} & & \\ & V = \left\{ \frac{-145}{62} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x-\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-60& = & 21x-315 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-21x} & = & \color{red}{21x} -315 \color{blue}{-21x} \color{blue}{+60} \\\Leftrightarrow & 14x& = & -255 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{-255}{14} \\\Leftrightarrow & x = \frac{-255}{14} & & \\ & V = \left\{ \frac{-255}{14} \right\} & \\\end{align}\)
10. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{140}{ \color{blue}{40} }x-\frac{280}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+25& = & 140x-280 \\\Leftrightarrow & 8x \color{red}{+25} \color{blue}{-25} \color{blue}{-140x} & = & \color{red}{140x} -280 \color{blue}{-140x} \color{blue}{-25} \\\Leftrightarrow & -132x& = & -305 \\\Leftrightarrow & \frac{-132x}{ \color{red}{-132} }& = & \frac{-305}{-132} \\\Leftrightarrow & x = \frac{305}{132} & & \\ & V = \left\{ \frac{305}{132} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -12x-60 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+12x} & = & \color{red}{-12x} -60 \color{blue}{+12x} \color{blue}{+8} \\\Leftrightarrow & 17x& = & -52 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-52}{17} \\\Leftrightarrow & x = \frac{-52}{17} & & \\ & V = \left\{ \frac{-52}{17} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -70x+294 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+70x} & = & \color{red}{-70x} +294 \color{blue}{+70x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & 270 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{270}{91} \\\Leftrightarrow & x = \frac{270}{91} & & \\ & V = \left\{ \frac{270}{91} \right\} & \\\end{align}\)