Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 112x+588 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-112x} & = & \color{red}{112x} +588 \color{blue}{-112x} \color{blue}{+36} \\\Leftrightarrow & -91x& = & 624 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{624}{-91} \\\Leftrightarrow & x = \frac{-48}{7} & & \\ & V = \left\{ \frac{-48}{7} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{7}& = & \frac{3}{2}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-30& = & 63x+168 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-63x} & = & \color{red}{63x} +168 \color{blue}{-63x} \color{blue}{+30} \\\Leftrightarrow & -49x& = & 198 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{198}{-49} \\\Leftrightarrow & x = \frac{-198}{49} & & \\ & V = \left\{ \frac{-198}{49} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-48& = & -63x+84 \\\Leftrightarrow & 28x \color{red}{-48} \color{blue}{+48} \color{blue}{+63x} & = & \color{red}{-63x} +84 \color{blue}{+63x} \color{blue}{+48} \\\Leftrightarrow & 91x& = & 132 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{132}{91} \\\Leftrightarrow & x = \frac{132}{91} & & \\ & V = \left\{ \frac{132}{91} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 252x+1050 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-252x} & = & \color{red}{252x} +1050 \color{blue}{-252x} \color{blue}{-150} \\\Leftrightarrow & -217x& = & 900 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{900}{-217} \\\Leftrightarrow & x = \frac{-900}{217} & & \\ & V = \left\{ \frac{-900}{217} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{5}{4}x+3 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{195}{ \color{blue}{156} }x+\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+48& = & 195x+468 \\\Leftrightarrow & 52x \color{red}{+48} \color{blue}{-48} \color{blue}{-195x} & = & \color{red}{195x} +468 \color{blue}{-195x} \color{blue}{-48} \\\Leftrightarrow & -143x& = & 420 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{420}{-143} \\\Leftrightarrow & x = \frac{-420}{143} & & \\ & V = \left\{ \frac{-420}{143} \right\} & \\\end{align}\)
6. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{1}{4}x-2 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 140 }{ \color{blue}{308} })& = & (\frac{77}{ \color{blue}{308} }x-\frac{616}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+140& = & 77x-616 \\\Leftrightarrow & 44x \color{red}{+140} \color{blue}{-140} \color{blue}{-77x} & = & \color{red}{77x} -616 \color{blue}{-77x} \color{blue}{-140} \\\Leftrightarrow & -33x& = & -756 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-756}{-33} \\\Leftrightarrow & x = \frac{252}{11} & & \\ & V = \left\{ \frac{252}{11} \right\} & \\\end{align}\)
7. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 168 }{ \color{blue}{546} })& = & (\frac{-455}{ \color{blue}{546} }x+\frac{3276}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-168& = & -455x+3276 \\\Leftrightarrow & 78x \color{red}{-168} \color{blue}{+168} \color{blue}{+455x} & = & \color{red}{-455x} +3276 \color{blue}{+455x} \color{blue}{+168} \\\Leftrightarrow & 533x& = & 3444 \\\Leftrightarrow & \frac{533x}{ \color{red}{533} }& = & \frac{3444}{533} \\\Leftrightarrow & x = \frac{84}{13} & & \\ & V = \left\{ \frac{84}{13} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & -132x-990 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{+132x} & = & \color{red}{-132x} -990 \color{blue}{+132x} \color{blue}{-90} \\\Leftrightarrow & 187x& = & -1080 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-1080}{187} \\\Leftrightarrow & x = \frac{-1080}{187} & & \\ & V = \left\{ \frac{-1080}{187} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -12x-60 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+12x} & = & \color{red}{-12x} -60 \color{blue}{+12x} \color{blue}{+4} \\\Leftrightarrow & 17x& = & -56 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-56}{17} \\\Leftrightarrow & x = \frac{-56}{17} & & \\ & V = \left\{ \frac{-56}{17} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & -50x-60 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{+50x} & = & \color{red}{-50x} -60 \color{blue}{+50x} \color{blue}{+9} \\\Leftrightarrow & 65x& = & -51 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{-51}{65} \\\Leftrightarrow & x = \frac{-51}{65} & & \\ & V = \left\{ \frac{-51}{65} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & -312x+780 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{+312x} & = & \color{red}{-312x} +780 \color{blue}{+312x} \color{blue}{+90} \\\Leftrightarrow & 377x& = & 870 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{870}{377} \\\Leftrightarrow & x = \frac{30}{13} & & \\ & V = \left\{ \frac{30}{13} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & -175x+1260 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{+175x} & = & \color{red}{-175x} +1260 \color{blue}{+175x} \color{blue}{-28} \\\Leftrightarrow & 205x& = & 1232 \\\Leftrightarrow & \frac{205x}{ \color{red}{205} }& = & \frac{1232}{205} \\\Leftrightarrow & x = \frac{1232}{205} & & \\ & V = \left\{ \frac{1232}{205} \right\} & \\\end{align}\)