Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 252x+1050 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-252x} & = & \color{red}{252x} +1050 \color{blue}{-252x} \color{blue}{-150} \\\Leftrightarrow & -217x& = & 900 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{900}{-217} \\\Leftrightarrow & x = \frac{-900}{217} & & \\ & V = \left\{ \frac{-900}{217} \right\} & \\\end{align}\)
2. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 70 }{ \color{blue}{385} })& = & (\frac{-308}{ \color{blue}{385} }x-\frac{770}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-70& = & -308x-770 \\\Leftrightarrow & 55x \color{red}{-70} \color{blue}{+70} \color{blue}{+308x} & = & \color{red}{-308x} -770 \color{blue}{+308x} \color{blue}{+70} \\\Leftrightarrow & 363x& = & -700 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{-700}{363} \\\Leftrightarrow & x = \frac{-700}{363} & & \\ & V = \left\{ \frac{-700}{363} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{5}{2}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+4& = & 75x+150 \\\Leftrightarrow & 10x \color{red}{+4} \color{blue}{-4} \color{blue}{-75x} & = & \color{red}{75x} +150 \color{blue}{-75x} \color{blue}{-4} \\\Leftrightarrow & -65x& = & 146 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{146}{-65} \\\Leftrightarrow & x = \frac{-146}{65} & & \\ & V = \left\{ \frac{-146}{65} \right\} & \\\end{align}\)
4. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 105 }{ \color{blue}{273} })& = & (\frac{364}{ \color{blue}{273} }x+\frac{1092}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-105& = & 364x+1092 \\\Leftrightarrow & 39x \color{red}{-105} \color{blue}{+105} \color{blue}{-364x} & = & \color{red}{364x} +1092 \color{blue}{-364x} \color{blue}{+105} \\\Leftrightarrow & -325x& = & 1197 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{1197}{-325} \\\Leftrightarrow & x = \frac{-1197}{325} & & \\ & V = \left\{ \frac{-1197}{325} \right\} & \\\end{align}\)
5. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{2}{5}x-7 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{8}{ \color{blue}{20} }x-\frac{140}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & 8x-140 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{-8x} & = & \color{red}{8x} -140 \color{blue}{-8x} \color{blue}{-6} \\\Leftrightarrow & -3x& = & -146 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-146}{-3} \\\Leftrightarrow & x = \frac{146}{3} & & \\ & V = \left\{ \frac{146}{3} \right\} & \\\end{align}\)
6. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-18}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & -18x+192 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{+18x} & = & \color{red}{-18x} +192 \color{blue}{+18x} \color{blue}{+15} \\\Leftrightarrow & 26x& = & 207 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{207}{26} \\\Leftrightarrow & x = \frac{207}{26} & & \\ & V = \left\{ \frac{207}{26} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x+156 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} +156 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & 144 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{144}{-13} \\\Leftrightarrow & x = \frac{-144}{13} & & \\ & V = \left\{ \frac{-144}{13} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & -60x+108 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+60x} & = & \color{red}{-60x} +108 \color{blue}{+60x} \color{blue}{+8} \\\Leftrightarrow & 69x& = & 116 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{116}{69} \\\Leftrightarrow & x = \frac{116}{69} & & \\ & V = \left\{ \frac{116}{69} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+40& = & -105x-420 \\\Leftrightarrow & 28x \color{red}{+40} \color{blue}{-40} \color{blue}{+105x} & = & \color{red}{-105x} -420 \color{blue}{+105x} \color{blue}{-40} \\\Leftrightarrow & 133x& = & -460 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-460}{133} \\\Leftrightarrow & x = \frac{-460}{133} & & \\ & V = \left\{ \frac{-460}{133} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x-\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-9& = & -128x-288 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{+128x} & = & \color{red}{-128x} -288 \color{blue}{+128x} \color{blue}{+9} \\\Leftrightarrow & 140x& = & -279 \\\Leftrightarrow & \frac{140x}{ \color{red}{140} }& = & \frac{-279}{140} \\\Leftrightarrow & x = \frac{-279}{140} & & \\ & V = \left\{ \frac{-279}{140} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & -112x-42 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+112x} & = & \color{red}{-112x} -42 \color{blue}{+112x} \color{blue}{+30} \\\Leftrightarrow & 133x& = & -12 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-12}{133} \\\Leftrightarrow & x = \frac{-12}{133} & & \\ & V = \left\{ \frac{-12}{133} \right\} & \\\end{align}\)
12. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{5}{6}x+1 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 84 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x+\frac{546}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-84& = & 455x+546 \\\Leftrightarrow & 78x \color{red}{-84} \color{blue}{+84} \color{blue}{-455x} & = & \color{red}{455x} +546 \color{blue}{-455x} \color{blue}{+84} \\\Leftrightarrow & -377x& = & 630 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{630}{-377} \\\Leftrightarrow & x = \frac{-630}{377} & & \\ & V = \left\{ \frac{-630}{377} \right\} & \\\end{align}\)