Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x-56 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} -56 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & -50 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-50}{-5} \\\Leftrightarrow & x = 10 & & \\ & V = \left\{ 10 \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -294x+840 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+294x} & = & \color{red}{-294x} +840 \color{blue}{+294x} \color{blue}{+60} \\\Leftrightarrow & 329x& = & 900 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{900}{329} \\\Leftrightarrow & x = \frac{900}{329} & & \\ & V = \left\{ \frac{900}{329} \right\} & \\\end{align}\)
3. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{5}{4}x+8 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 8 }{ \color{blue}{28} })& = & (\frac{35}{ \color{blue}{28} }x+\frac{224}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-8& = & 35x+224 \\\Leftrightarrow & 4x \color{red}{-8} \color{blue}{+8} \color{blue}{-35x} & = & \color{red}{35x} +224 \color{blue}{-35x} \color{blue}{+8} \\\Leftrightarrow & -31x& = & 232 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{232}{-31} \\\Leftrightarrow & x = \frac{-232}{31} & & \\ & V = \left\{ \frac{-232}{31} \right\} & \\\end{align}\)
4. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{315.} (\frac{45x}{ \color{blue}{315} }+ \frac{ 70 }{ \color{blue}{315} })& = & (\frac{-252}{ \color{blue}{315} }x+\frac{630}{ \color{blue}{315} }) \color{blue}{.315} \\\Leftrightarrow & 45x+70& = & -252x+630 \\\Leftrightarrow & 45x \color{red}{+70} \color{blue}{-70} \color{blue}{+252x} & = & \color{red}{-252x} +630 \color{blue}{+252x} \color{blue}{-70} \\\Leftrightarrow & 297x& = & 560 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{560}{297} \\\Leftrightarrow & x = \frac{560}{297} & & \\ & V = \left\{ \frac{560}{297} \right\} & \\\end{align}\)
5. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -112x+840 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+112x} & = & \color{red}{-112x} +840 \color{blue}{+112x} \color{blue}{-80} \\\Leftrightarrow & 147x& = & 760 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{760}{147} \\\Leftrightarrow & x = \frac{760}{147} & & \\ & V = \left\{ \frac{760}{147} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-16& = & -27x+108 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{+27x} & = & \color{red}{-27x} +108 \color{blue}{+27x} \color{blue}{+16} \\\Leftrightarrow & 39x& = & 124 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{124}{39} \\\Leftrightarrow & x = \frac{124}{39} & & \\ & V = \left\{ \frac{124}{39} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x+\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-10& = & -48x+36 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{+48x} & = & \color{red}{-48x} +36 \color{blue}{+48x} \color{blue}{+10} \\\Leftrightarrow & 57x& = & 46 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{46}{57} \\\Leftrightarrow & x = \frac{46}{57} & & \\ & V = \left\{ \frac{46}{57} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & -196x+700 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+196x} & = & \color{red}{-196x} +700 \color{blue}{+196x} \color{blue}{+40} \\\Leftrightarrow & 231x& = & 740 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{740}{231} \\\Leftrightarrow & x = \frac{740}{231} & & \\ & V = \left\{ \frac{740}{231} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -56x+350 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+56x} & = & \color{red}{-56x} +350 \color{blue}{+56x} \color{blue}{+20} \\\Leftrightarrow & 91x& = & 370 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{370}{91} \\\Leftrightarrow & x = \frac{370}{91} & & \\ & V = \left\{ \frac{370}{91} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & -52x+78 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+52x} & = & \color{red}{-52x} +78 \color{blue}{+52x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & 102 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{102}{91} \\\Leftrightarrow & x = \frac{102}{91} & & \\ & V = \left\{ \frac{102}{91} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{-7}{4}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & -147x-672 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{+147x} & = & \color{red}{-147x} -672 \color{blue}{+147x} \color{blue}{-48} \\\Leftrightarrow & 175x& = & -720 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-720}{175} \\\Leftrightarrow & x = \frac{-144}{35} & & \\ & V = \left\{ \frac{-144}{35} \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & -44x+528 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{+44x} & = & \color{red}{-44x} +528 \color{blue}{+44x} \color{blue}{+12} \\\Leftrightarrow & 77x& = & 540 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{540}{77} \\\Leftrightarrow & x = \frac{540}{77} & & \\ & V = \left\{ \frac{540}{77} \right\} & \\\end{align}\)