Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-32}{ \color{blue}{80} }x+\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & -32x+240 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{+32x} & = & \color{red}{-32x} +240 \color{blue}{+32x} \color{blue}{+25} \\\Leftrightarrow & 52x& = & 265 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{265}{52} \\\Leftrightarrow & x = \frac{265}{52} & & \\ & V = \left\{ \frac{265}{52} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-30& = & 33x-66 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-33x} & = & \color{red}{33x} -66 \color{blue}{-33x} \color{blue}{+30} \\\Leftrightarrow & -11x& = & -36 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-36}{-11} \\\Leftrightarrow & x = \frac{36}{11} & & \\ & V = \left\{ \frac{36}{11} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & -52x-390 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+52x} & = & \color{red}{-52x} -390 \color{blue}{+52x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -414 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-414}{91} \\\Leftrightarrow & x = \frac{-414}{91} & & \\ & V = \left\{ \frac{-414}{91} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & -462x-2640 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{+462x} & = & \color{red}{-462x} -2640 \color{blue}{+462x} \color{blue}{-150} \\\Leftrightarrow & 517x& = & -2790 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-2790}{517} \\\Leftrightarrow & x = \frac{-2790}{517} & & \\ & V = \left\{ \frac{-2790}{517} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 196x+168 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-196x} & = & \color{red}{196x} +168 \color{blue}{-196x} \color{blue}{-48} \\\Leftrightarrow & -175x& = & 120 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{120}{-175} \\\Leftrightarrow & x = \frac{-24}{35} & & \\ & V = \left\{ \frac{-24}{35} \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 42 }{ \color{blue}{154} })& = & (\frac{539}{ \color{blue}{154} }x+\frac{1232}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-42& = & 539x+1232 \\\Leftrightarrow & 22x \color{red}{-42} \color{blue}{+42} \color{blue}{-539x} & = & \color{red}{539x} +1232 \color{blue}{-539x} \color{blue}{+42} \\\Leftrightarrow & -517x& = & 1274 \\\Leftrightarrow & \frac{-517x}{ \color{red}{-517} }& = & \frac{1274}{-517} \\\Leftrightarrow & x = \frac{-1274}{517} & & \\ & V = \left\{ \frac{-1274}{517} \right\} & \\\end{align}\)
7. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 28 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-28& = & 77x-1078 \\\Leftrightarrow & 22x \color{red}{-28} \color{blue}{+28} \color{blue}{-77x} & = & \color{red}{77x} -1078 \color{blue}{-77x} \color{blue}{+28} \\\Leftrightarrow & -55x& = & -1050 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-1050}{-55} \\\Leftrightarrow & x = \frac{210}{11} & & \\ & V = \left\{ \frac{210}{11} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{2}{3}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 63 }{ \color{blue}{210} })& = & (\frac{140}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+63& = & 140x-840 \\\Leftrightarrow & 30x \color{red}{+63} \color{blue}{-63} \color{blue}{-140x} & = & \color{red}{140x} -840 \color{blue}{-140x} \color{blue}{-63} \\\Leftrightarrow & -110x& = & -903 \\\Leftrightarrow & \frac{-110x}{ \color{red}{-110} }& = & \frac{-903}{-110} \\\Leftrightarrow & x = \frac{903}{110} & & \\ & V = \left\{ \frac{903}{110} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{11}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-90& = & -462x+990 \\\Leftrightarrow & 55x \color{red}{-90} \color{blue}{+90} \color{blue}{+462x} & = & \color{red}{-462x} +990 \color{blue}{+462x} \color{blue}{+90} \\\Leftrightarrow & 517x& = & 1080 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{1080}{517} \\\Leftrightarrow & x = \frac{1080}{517} & & \\ & V = \left\{ \frac{1080}{517} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 8 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x+\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+8& = & 49x+112 \\\Leftrightarrow & 2x \color{red}{+8} \color{blue}{-8} \color{blue}{-49x} & = & \color{red}{49x} +112 \color{blue}{-49x} \color{blue}{-8} \\\Leftrightarrow & -47x& = & 104 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{104}{-47} \\\Leftrightarrow & x = \frac{-104}{47} & & \\ & V = \left\{ \frac{-104}{47} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 15x+30 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} +30 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -9x& = & 34 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{34}{-9} \\\Leftrightarrow & x = \frac{-34}{9} & & \\ & V = \left\{ \frac{-34}{9} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x-1050 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} -1050 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & -990 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-990}{203} \\\Leftrightarrow & x = \frac{-990}{203} & & \\ & V = \left\{ \frac{-990}{203} \right\} & \\\end{align}\)