Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 84 }{ \color{blue}{273} })& = & (\frac{-455}{ \color{blue}{273} }x+\frac{1638}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-84& = & -455x+1638 \\\Leftrightarrow & 39x \color{red}{-84} \color{blue}{+84} \color{blue}{+455x} & = & \color{red}{-455x} +1638 \color{blue}{+455x} \color{blue}{+84} \\\Leftrightarrow & 494x& = & 1722 \\\Leftrightarrow & \frac{494x}{ \color{red}{494} }& = & \frac{1722}{494} \\\Leftrightarrow & x = \frac{861}{247} & & \\ & V = \left\{ \frac{861}{247} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{7}{6}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{455}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 455x-3120 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-455x} & = & \color{red}{455x} -3120 \color{blue}{-455x} \color{blue}{-60} \\\Leftrightarrow & -377x& = & -3180 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{-3180}{-377} \\\Leftrightarrow & x = \frac{3180}{377} & & \\ & V = \left\{ \frac{3180}{377} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 168x-1050 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-168x} & = & \color{red}{168x} -1050 \color{blue}{-168x} \color{blue}{+60} \\\Leftrightarrow & -133x& = & -990 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-990}{-133} \\\Leftrightarrow & x = \frac{990}{133} & & \\ & V = \left\{ \frac{990}{133} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x-192 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} -192 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -201 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-201}{-8} \\\Leftrightarrow & x = \frac{201}{8} & & \\ & V = \left\{ \frac{201}{8} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 20x+240 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-20x} & = & \color{red}{20x} +240 \color{blue}{-20x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & 232 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{232}{-5} \\\Leftrightarrow & x = \frac{-232}{5} & & \\ & V = \left\{ \frac{-232}{5} \right\} & \\\end{align}\)
6. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{7}{2}x+2 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{140}{ \color{blue}{40} }x+\frac{80}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+25& = & 140x+80 \\\Leftrightarrow & 8x \color{red}{+25} \color{blue}{-25} \color{blue}{-140x} & = & \color{red}{140x} +80 \color{blue}{-140x} \color{blue}{-25} \\\Leftrightarrow & -132x& = & 55 \\\Leftrightarrow & \frac{-132x}{ \color{red}{-132} }& = & \frac{55}{-132} \\\Leftrightarrow & x = \frac{-5}{12} & & \\ & V = \left\{ \frac{-5}{12} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & -196x-840 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+196x} & = & \color{red}{-196x} -840 \color{blue}{+196x} \color{blue}{+40} \\\Leftrightarrow & 231x& = & -800 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-800}{231} \\\Leftrightarrow & x = \frac{-800}{231} & & \\ & V = \left\{ \frac{-800}{231} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -112x-140 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+112x} & = & \color{red}{-112x} -140 \color{blue}{+112x} \color{blue}{-80} \\\Leftrightarrow & 147x& = & -220 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-220}{147} \\\Leftrightarrow & x = \frac{-220}{147} & & \\ & V = \left\{ \frac{-220}{147} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-60& = & -196x+560 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+196x} & = & \color{red}{-196x} +560 \color{blue}{+196x} \color{blue}{+60} \\\Leftrightarrow & 231x& = & 620 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{620}{231} \\\Leftrightarrow & x = \frac{620}{231} & & \\ & V = \left\{ \frac{620}{231} \right\} & \\\end{align}\)
10. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{4}{3}x+3 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{140}{ \color{blue}{105} }x+\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+60& = & 140x+315 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-140x} & = & \color{red}{140x} +315 \color{blue}{-140x} \color{blue}{-60} \\\Leftrightarrow & -119x& = & 255 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{255}{-119} \\\Leftrightarrow & x = \frac{-15}{7} & & \\ & V = \left\{ \frac{-15}{7} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x+\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & -12x+144 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} +144 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 21x& = & 140 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{140}{21} \\\Leftrightarrow & x = \frac{20}{3} & & \\ & V = \left\{ \frac{20}{3} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{182}{ \color{blue}{78} }x+\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-30& = & 182x+234 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{-182x} & = & \color{red}{182x} +234 \color{blue}{-182x} \color{blue}{+30} \\\Leftrightarrow & -143x& = & 264 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{264}{-143} \\\Leftrightarrow & x = \frac{-24}{13} & & \\ & V = \left\{ \frac{-24}{13} \right\} & \\\end{align}\)