Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{-3}{4}x+4 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x+\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+48& = & -99x+528 \\\Leftrightarrow & 44x \color{red}{+48} \color{blue}{-48} \color{blue}{+99x} & = & \color{red}{-99x} +528 \color{blue}{+99x} \color{blue}{-48} \\\Leftrightarrow & 143x& = & 480 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{480}{143} \\\Leftrightarrow & x = \frac{480}{143} & & \\ & V = \left\{ \frac{480}{143} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x+98 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} +98 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 102 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{102}{-5} \\\Leftrightarrow & x = \frac{-102}{5} & & \\ & V = \left\{ \frac{-102}{5} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & 6x+126 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-6x} & = & \color{red}{6x} +126 \color{blue}{-6x} \color{blue}{-8} \\\Leftrightarrow & 3x& = & 118 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{118}{3} \\\Leftrightarrow & x = \frac{118}{3} & & \\ & V = \left\{ \frac{118}{3} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & -98x-210 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+98x} & = & \color{red}{-98x} -210 \color{blue}{+98x} \color{blue}{+40} \\\Leftrightarrow & 133x& = & -170 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-170}{133} \\\Leftrightarrow & x = \frac{-170}{133} & & \\ & V = \left\{ \frac{-170}{133} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & -128x+192 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{+128x} & = & \color{red}{-128x} +192 \color{blue}{+128x} \color{blue}{-9} \\\Leftrightarrow & 152x& = & 183 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{183}{152} \\\Leftrightarrow & x = \frac{183}{152} & & \\ & V = \left\{ \frac{183}{152} \right\} & \\\end{align}\)
6. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{13}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 175 }{ \color{blue}{455} })& = & (\frac{-182}{ \color{blue}{455} }x-\frac{1820}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+175& = & -182x-1820 \\\Leftrightarrow & 65x \color{red}{+175} \color{blue}{-175} \color{blue}{+182x} & = & \color{red}{-182x} -1820 \color{blue}{+182x} \color{blue}{-175} \\\Leftrightarrow & 247x& = & -1995 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-1995}{247} \\\Leftrightarrow & x = \frac{-105}{13} & & \\ & V = \left\{ \frac{-105}{13} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -168x+1260 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+168x} & = & \color{red}{-168x} +1260 \color{blue}{+168x} \color{blue}{-90} \\\Leftrightarrow & 203x& = & 1170 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1170}{203} \\\Leftrightarrow & x = \frac{1170}{203} & & \\ & V = \left\{ \frac{1170}{203} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{110}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 110x-66 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-110x} & = & \color{red}{110x} -66 \color{blue}{-110x} \color{blue}{-12} \\\Leftrightarrow & -77x& = & -78 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-78}{-77} \\\Leftrightarrow & x = \frac{78}{77} & & \\ & V = \left\{ \frac{78}{77} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x+420 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} +420 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & 380 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{380}{-21} \\\Leftrightarrow & x = \frac{-380}{21} & & \\ & V = \left\{ \frac{-380}{21} \right\} & \\\end{align}\)
10. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-325}{ \color{blue}{195} }x-\frac{195}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+45& = & -325x-195 \\\Leftrightarrow & 39x \color{red}{+45} \color{blue}{-45} \color{blue}{+325x} & = & \color{red}{-325x} -195 \color{blue}{+325x} \color{blue}{-45} \\\Leftrightarrow & 364x& = & -240 \\\Leftrightarrow & \frac{364x}{ \color{red}{364} }& = & \frac{-240}{364} \\\Leftrightarrow & x = \frac{-60}{91} & & \\ & V = \left\{ \frac{-60}{91} \right\} & \\\end{align}\)
11. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }- \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-8}{ \color{blue}{20} }x-\frac{140}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x-6& = & -8x-140 \\\Leftrightarrow & 5x \color{red}{-6} \color{blue}{+6} \color{blue}{+8x} & = & \color{red}{-8x} -140 \color{blue}{+8x} \color{blue}{+6} \\\Leftrightarrow & 13x& = & -134 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-134}{13} \\\Leftrightarrow & x = \frac{-134}{13} & & \\ & V = \left\{ \frac{-134}{13} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-18}{ \color{blue}{24} }x-\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & -18x-168 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{+18x} & = & \color{red}{-18x} -168 \color{blue}{+18x} \color{blue}{-9} \\\Leftrightarrow & 26x& = & -177 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-177}{26} \\\Leftrightarrow & x = \frac{-177}{26} & & \\ & V = \left\{ \frac{-177}{26} \right\} & \\\end{align}\)