Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x-\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & -275x-2310 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{+275x} & = & \color{red}{-275x} -2310 \color{blue}{+275x} \color{blue}{-150} \\\Leftrightarrow & 341x& = & -2460 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{-2460}{341} \\\Leftrightarrow & x = \frac{-2460}{341} & & \\ & V = \left\{ \frac{-2460}{341} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-24& = & -88x+792 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+88x} & = & \color{red}{-88x} +792 \color{blue}{+88x} \color{blue}{+24} \\\Leftrightarrow & 121x& = & 816 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{816}{121} \\\Leftrightarrow & x = \frac{816}{121} & & \\ & V = \left\{ \frac{816}{121} \right\} & \\\end{align}\)
3. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 70 }{ \color{blue}{385} })& = & (\frac{462}{ \color{blue}{385} }x+\frac{1540}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+70& = & 462x+1540 \\\Leftrightarrow & 55x \color{red}{+70} \color{blue}{-70} \color{blue}{-462x} & = & \color{red}{462x} +1540 \color{blue}{-462x} \color{blue}{-70} \\\Leftrightarrow & -407x& = & 1470 \\\Leftrightarrow & \frac{-407x}{ \color{red}{-407} }& = & \frac{1470}{-407} \\\Leftrightarrow & x = \frac{-1470}{407} & & \\ & V = \left\{ \frac{-1470}{407} \right\} & \\\end{align}\)
4. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 28 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{1456}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-28& = & 91x+1456 \\\Leftrightarrow & 26x \color{red}{-28} \color{blue}{+28} \color{blue}{-91x} & = & \color{red}{91x} +1456 \color{blue}{-91x} \color{blue}{+28} \\\Leftrightarrow & -65x& = & 1484 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1484}{-65} \\\Leftrightarrow & x = \frac{-1484}{65} & & \\ & V = \left\{ \frac{-1484}{65} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & -264x-990 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{+264x} & = & \color{red}{-264x} -990 \color{blue}{+264x} \color{blue}{+120} \\\Leftrightarrow & 319x& = & -870 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-870}{319} \\\Leftrightarrow & x = \frac{-30}{11} & & \\ & V = \left\{ \frac{-30}{11} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -12x-210 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+12x} & = & \color{red}{-12x} -210 \color{blue}{+12x} \color{blue}{+9} \\\Leftrightarrow & 17x& = & -201 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-201}{17} \\\Leftrightarrow & x = \frac{-201}{17} & & \\ & V = \left\{ \frac{-201}{17} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{225}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 225x+270 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-225x} & = & \color{red}{225x} +270 \color{blue}{-225x} \color{blue}{+40} \\\Leftrightarrow & -207x& = & 310 \\\Leftrightarrow & \frac{-207x}{ \color{red}{-207} }& = & \frac{310}{-207} \\\Leftrightarrow & x = \frac{-310}{207} & & \\ & V = \left\{ \frac{-310}{207} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -126x+360 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+126x} & = & \color{red}{-126x} +360 \color{blue}{+126x} \color{blue}{+40} \\\Leftrightarrow & 141x& = & 400 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{400}{141} \\\Leftrightarrow & x = \frac{400}{141} & & \\ & V = \left\{ \frac{400}{141} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{364}{ \color{blue}{156} }x+\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 364x+1248 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-364x} & = & \color{red}{364x} +1248 \color{blue}{-364x} \color{blue}{+36} \\\Leftrightarrow & -325x& = & 1284 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{1284}{-325} \\\Leftrightarrow & x = \frac{-1284}{325} & & \\ & V = \left\{ \frac{-1284}{325} \right\} & \\\end{align}\)
10. \(\text{560 is het kleinste gemene veelvoud van 7, 16 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{560.} (\frac{80x}{ \color{blue}{560} }+ \frac{ 175 }{ \color{blue}{560} })& = & (\frac{112}{ \color{blue}{560} }x-\frac{560}{ \color{blue}{560} }) \color{blue}{.560} \\\Leftrightarrow & 80x+175& = & 112x-560 \\\Leftrightarrow & 80x \color{red}{+175} \color{blue}{-175} \color{blue}{-112x} & = & \color{red}{112x} -560 \color{blue}{-112x} \color{blue}{-175} \\\Leftrightarrow & -32x& = & -735 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{-735}{-32} \\\Leftrightarrow & x = \frac{735}{32} & & \\ & V = \left\{ \frac{735}{32} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{195}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 195x-312 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-195x} & = & \color{red}{195x} -312 \color{blue}{-195x} \color{blue}{+24} \\\Leftrightarrow & -169x& = & -288 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-288}{-169} \\\Leftrightarrow & x = \frac{288}{169} & & \\ & V = \left\{ \frac{288}{169} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{7}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+9& = & 147x-168 \\\Leftrightarrow & 14x \color{red}{+9} \color{blue}{-9} \color{blue}{-147x} & = & \color{red}{147x} -168 \color{blue}{-147x} \color{blue}{-9} \\\Leftrightarrow & -133x& = & -177 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-177}{-133} \\\Leftrightarrow & x = \frac{177}{133} & & \\ & V = \left\{ \frac{177}{133} \right\} & \\\end{align}\)