Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{560 is het kleinste gemene veelvoud van 7, 16 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{560.} (\frac{80x}{ \color{blue}{560} }+ \frac{ 105 }{ \color{blue}{560} })& = & (\frac{-784}{ \color{blue}{560} }x-\frac{560}{ \color{blue}{560} }) \color{blue}{.560} \\\Leftrightarrow & 80x+105& = & -784x-560 \\\Leftrightarrow & 80x \color{red}{+105} \color{blue}{-105} \color{blue}{+784x} & = & \color{red}{-784x} -560 \color{blue}{+784x} \color{blue}{-105} \\\Leftrightarrow & 864x& = & -665 \\\Leftrightarrow & \frac{864x}{ \color{red}{864} }& = & \frac{-665}{864} \\\Leftrightarrow & x = \frac{-665}{864} & & \\ & V = \left\{ \frac{-665}{864} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{3}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 234x+1560 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-234x} & = & \color{red}{234x} +1560 \color{blue}{-234x} \color{blue}{-90} \\\Leftrightarrow & -169x& = & 1470 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{1470}{-169} \\\Leftrightarrow & x = \frac{-1470}{169} & & \\ & V = \left\{ \frac{-1470}{169} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & -63x+84 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{+63x} & = & \color{red}{-63x} +84 \color{blue}{+63x} \color{blue}{+36} \\\Leftrightarrow & 91x& = & 120 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{120}{91} \\\Leftrightarrow & x = \frac{120}{91} & & \\ & V = \left\{ \frac{120}{91} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{182}{ \color{blue}{78} }x+\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 182x+546 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-182x} & = & \color{red}{182x} +546 \color{blue}{-182x} \color{blue}{-12} \\\Leftrightarrow & -143x& = & 534 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{534}{-143} \\\Leftrightarrow & x = \frac{-534}{143} & & \\ & V = \left\{ \frac{-534}{143} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x-\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & -60x-288 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{+60x} & = & \color{red}{-60x} -288 \color{blue}{+60x} \color{blue}{-16} \\\Leftrightarrow & 69x& = & -304 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{-304}{69} \\\Leftrightarrow & x = \frac{-304}{69} & & \\ & V = \left\{ \frac{-304}{69} \right\} & \\\end{align}\)
6. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{-49}{ \color{blue}{35} }x+\frac{210}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & -49x+210 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{+49x} & = & \color{red}{-49x} +210 \color{blue}{+49x} \color{blue}{+10} \\\Leftrightarrow & 54x& = & 220 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{220}{54} \\\Leftrightarrow & x = \frac{110}{27} & & \\ & V = \left\{ \frac{110}{27} \right\} & \\\end{align}\)
7. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x+\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -192x+480 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+192x} & = & \color{red}{-192x} +480 \color{blue}{+192x} \color{blue}{-75} \\\Leftrightarrow & 232x& = & 405 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{405}{232} \\\Leftrightarrow & x = \frac{405}{232} & & \\ & V = \left\{ \frac{405}{232} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & 168x-980 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{-168x} & = & \color{red}{168x} -980 \color{blue}{-168x} \color{blue}{+80} \\\Leftrightarrow & -133x& = & -900 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-900}{-133} \\\Leftrightarrow & x = \frac{900}{133} & & \\ & V = \left\{ \frac{900}{133} \right\} & \\\end{align}\)
9. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{-7}{4}x-8 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 56 }{ \color{blue}{308} })& = & (\frac{-539}{ \color{blue}{308} }x-\frac{2464}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+56& = & -539x-2464 \\\Leftrightarrow & 44x \color{red}{+56} \color{blue}{-56} \color{blue}{+539x} & = & \color{red}{-539x} -2464 \color{blue}{+539x} \color{blue}{-56} \\\Leftrightarrow & 583x& = & -2520 \\\Leftrightarrow & \frac{583x}{ \color{red}{583} }& = & \frac{-2520}{583} \\\Leftrightarrow & x = \frac{-2520}{583} & & \\ & V = \left\{ \frac{-2520}{583} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{2}{3}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{20}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & 20x+30 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{-20x} & = & \color{red}{20x} +30 \color{blue}{-20x} \color{blue}{+9} \\\Leftrightarrow & -5x& = & 39 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{39}{-5} \\\Leftrightarrow & x = \frac{-39}{5} & & \\ & V = \left\{ \frac{-39}{5} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{4}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 24x+120 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-24x} & = & \color{red}{24x} +120 \color{blue}{-24x} \color{blue}{+8} \\\Leftrightarrow & -19x& = & 128 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{128}{-19} \\\Leftrightarrow & x = \frac{-128}{19} & & \\ & V = \left\{ \frac{-128}{19} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{4}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & 24x-180 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{-24x} & = & \color{red}{24x} -180 \color{blue}{-24x} \color{blue}{-25} \\\Leftrightarrow & -19x& = & -205 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-205}{-19} \\\Leftrightarrow & x = \frac{205}{19} & & \\ & V = \left\{ \frac{205}{19} \right\} & \\\end{align}\)