Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 42 }{ \color{blue}{231} })& = & (\frac{-385}{ \color{blue}{231} }x-\frac{924}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+42& = & -385x-924 \\\Leftrightarrow & 33x \color{red}{+42} \color{blue}{-42} \color{blue}{+385x} & = & \color{red}{-385x} -924 \color{blue}{+385x} \color{blue}{-42} \\\Leftrightarrow & 418x& = & -966 \\\Leftrightarrow & \frac{418x}{ \color{red}{418} }& = & \frac{-966}{418} \\\Leftrightarrow & x = \frac{-483}{209} & & \\ & V = \left\{ \frac{-483}{209} \right\} & \\\end{align}\)
2. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{8}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{144}{ \color{blue}{120} }x-\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-45& = & 144x-480 \\\Leftrightarrow & 20x \color{red}{-45} \color{blue}{+45} \color{blue}{-144x} & = & \color{red}{144x} -480 \color{blue}{-144x} \color{blue}{+45} \\\Leftrightarrow & -124x& = & -435 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{-435}{-124} \\\Leftrightarrow & x = \frac{435}{124} & & \\ & V = \left\{ \frac{435}{124} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & -260x-156 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{+260x} & = & \color{red}{-260x} -156 \color{blue}{+260x} \color{blue}{+36} \\\Leftrightarrow & 299x& = & -120 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-120}{299} \\\Leftrightarrow & x = \frac{-120}{299} & & \\ & V = \left\{ \frac{-120}{299} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & -546x+1560 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{+546x} & = & \color{red}{-546x} +1560 \color{blue}{+546x} \color{blue}{-150} \\\Leftrightarrow & 611x& = & 1410 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{1410}{611} \\\Leftrightarrow & x = \frac{30}{13} & & \\ & V = \left\{ \frac{30}{13} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{2}{3}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{24}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & 24x+108 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-24x} & = & \color{red}{24x} +108 \color{blue}{-24x} \color{blue}{+8} \\\Leftrightarrow & -15x& = & 116 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{116}{-15} \\\Leftrightarrow & x = \frac{-116}{15} & & \\ & V = \left\{ \frac{-116}{15} \right\} & \\\end{align}\)
6. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{5}{3}x-6 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{275}{ \color{blue}{165} }x-\frac{990}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+75& = & 275x-990 \\\Leftrightarrow & 33x \color{red}{+75} \color{blue}{-75} \color{blue}{-275x} & = & \color{red}{275x} -990 \color{blue}{-275x} \color{blue}{-75} \\\Leftrightarrow & -242x& = & -1065 \\\Leftrightarrow & \frac{-242x}{ \color{red}{-242} }& = & \frac{-1065}{-242} \\\Leftrightarrow & x = \frac{1065}{242} & & \\ & V = \left\{ \frac{1065}{242} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 21x-168 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{-18} \\\Leftrightarrow & -7x& = & -186 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-186}{-7} \\\Leftrightarrow & x = \frac{186}{7} & & \\ & V = \left\{ \frac{186}{7} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+120& = & 65x+390 \\\Leftrightarrow & 78x \color{red}{+120} \color{blue}{-120} \color{blue}{-65x} & = & \color{red}{65x} +390 \color{blue}{-65x} \color{blue}{-120} \\\Leftrightarrow & 13x& = & 270 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{270}{13} \\\Leftrightarrow & x = \frac{270}{13} & & \\ & V = \left\{ \frac{270}{13} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & 196x-336 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-196x} & = & \color{red}{196x} -336 \color{blue}{-196x} \color{blue}{-60} \\\Leftrightarrow & -175x& = & -396 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-396}{-175} \\\Leftrightarrow & x = \frac{396}{175} & & \\ & V = \left\{ \frac{396}{175} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 12 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-24}{ \color{blue}{60} }x+\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-25& = & -24x+300 \\\Leftrightarrow & 15x \color{red}{-25} \color{blue}{+25} \color{blue}{+24x} & = & \color{red}{-24x} +300 \color{blue}{+24x} \color{blue}{+25} \\\Leftrightarrow & 39x& = & 325 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{325}{39} \\\Leftrightarrow & x = \frac{25}{3} & & \\ & V = \left\{ \frac{25}{3} \right\} & \\\end{align}\)
11. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{882}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 63x+882 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-63x} & = & \color{red}{63x} +882 \color{blue}{-63x} \color{blue}{-56} \\\Leftrightarrow & -45x& = & 826 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{826}{-45} \\\Leftrightarrow & x = \frac{-826}{45} & & \\ & V = \left\{ \frac{-826}{45} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{195}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-18& = & 195x-546 \\\Leftrightarrow & 26x \color{red}{-18} \color{blue}{+18} \color{blue}{-195x} & = & \color{red}{195x} -546 \color{blue}{-195x} \color{blue}{+18} \\\Leftrightarrow & -169x& = & -528 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-528}{-169} \\\Leftrightarrow & x = \frac{528}{169} & & \\ & V = \left\{ \frac{528}{169} \right\} & \\\end{align}\)