Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & 35x+252 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{-35x} & = & \color{red}{35x} +252 \color{blue}{-35x} \color{blue}{-24} \\\Leftrightarrow & -29x& = & 228 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{228}{-29} \\\Leftrightarrow & x = \frac{-228}{29} & & \\ & V = \left\{ \frac{-228}{29} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{4}{3}x-1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{48}{ \color{blue}{36} }x-\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & 48x-36 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{-48x} & = & \color{red}{48x} -36 \color{blue}{-48x} \color{blue}{-16} \\\Leftrightarrow & -39x& = & -52 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-52}{-39} \\\Leftrightarrow & x = \frac{4}{3} & & \\ & V = \left\{ \frac{4}{3} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{5}{6}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{325}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-90& = & 325x+3120 \\\Leftrightarrow & 78x \color{red}{-90} \color{blue}{+90} \color{blue}{-325x} & = & \color{red}{325x} +3120 \color{blue}{-325x} \color{blue}{+90} \\\Leftrightarrow & -247x& = & 3210 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{3210}{-247} \\\Leftrightarrow & x = \frac{-3210}{247} & & \\ & V = \left\{ \frac{-3210}{247} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-30& = & 33x+528 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-33x} & = & \color{red}{33x} +528 \color{blue}{-33x} \color{blue}{+30} \\\Leftrightarrow & -11x& = & 558 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{558}{-11} \\\Leftrightarrow & x = \frac{-558}{11} & & \\ & V = \left\{ \frac{-558}{11} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 112x-168 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-112x} & = & \color{red}{112x} -168 \color{blue}{-112x} \color{blue}{-24} \\\Leftrightarrow & -91x& = & -192 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-192}{-91} \\\Leftrightarrow & x = \frac{192}{91} & & \\ & V = \left\{ \frac{192}{91} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-10& = & 9x-18 \\\Leftrightarrow & 6x \color{red}{-10} \color{blue}{+10} \color{blue}{-9x} & = & \color{red}{9x} -18 \color{blue}{-9x} \color{blue}{+10} \\\Leftrightarrow & -3x& = & -8 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-8}{-3} \\\Leftrightarrow & x = \frac{8}{3} & & \\ & V = \left\{ \frac{8}{3} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 18 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+18& = & 196x+588 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{-196x} & = & \color{red}{196x} +588 \color{blue}{-196x} \color{blue}{-18} \\\Leftrightarrow & -175x& = & 570 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{570}{-175} \\\Leftrightarrow & x = \frac{-114}{35} & & \\ & V = \left\{ \frac{-114}{35} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 396x-660 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-396x} & = & \color{red}{396x} -660 \color{blue}{-396x} \color{blue}{-120} \\\Leftrightarrow & -341x& = & -780 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-780}{-341} \\\Leftrightarrow & x = \frac{780}{341} & & \\ & V = \left\{ \frac{780}{341} \right\} & \\\end{align}\)
9. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 112 }{ \color{blue}{420} })& = & (\frac{105}{ \color{blue}{420} }x+\frac{1260}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+112& = & 105x+1260 \\\Leftrightarrow & 60x \color{red}{+112} \color{blue}{-112} \color{blue}{-105x} & = & \color{red}{105x} +1260 \color{blue}{-105x} \color{blue}{-112} \\\Leftrightarrow & -45x& = & 1148 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{1148}{-45} \\\Leftrightarrow & x = \frac{-1148}{45} & & \\ & V = \left\{ \frac{-1148}{45} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-9}{ \color{blue}{12} }x-\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & -9x-72 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{+9x} & = & \color{red}{-9x} -72 \color{blue}{+9x} \color{blue}{-5} \\\Leftrightarrow & 13x& = & -77 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-77}{13} \\\Leftrightarrow & x = \frac{-77}{13} & & \\ & V = \left\{ \frac{-77}{13} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{3}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 126x+210 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-126x} & = & \color{red}{126x} +210 \color{blue}{-126x} \color{blue}{-60} \\\Leftrightarrow & -91x& = & 150 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{150}{-91} \\\Leftrightarrow & x = \frac{-150}{91} & & \\ & V = \left\{ \frac{-150}{91} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{7}{4}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{140}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 140x+560 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-140x} & = & \color{red}{140x} +560 \color{blue}{-140x} \color{blue}{-15} \\\Leftrightarrow & -124x& = & 545 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{545}{-124} \\\Leftrightarrow & x = \frac{-545}{124} & & \\ & V = \left\{ \frac{-545}{124} \right\} & \\\end{align}\)