Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{4}-\frac{5}{9}=\frac{2}{3}x-7\)
2. \(\frac{x}{6}-\frac{5}{14}=\frac{7}{5}x+3\)
3. \(\frac{x}{6}+\frac{4}{7}=\frac{3}{5}x+3\)
4. \(\frac{x}{2}+\frac{5}{9}=\frac{-5}{3}x+1\)
5. \(\frac{x}{2}+\frac{5}{8}=\frac{-5}{3}x-6\)
6. \(\frac{x}{7}+\frac{5}{7}=\frac{1}{2}x+4\)
7. \(\frac{x}{6}+\frac{3}{7}=\frac{-2}{5}x-7\)
8. \(\frac{x}{2}+\frac{4}{15}=\frac{7}{5}x+6\)
9. \(\frac{x}{4}+\frac{5}{11}=\frac{-5}{3}x-6\)
10. \(\frac{x}{6}-\frac{4}{9}=\frac{-7}{5}x-7\)
11. \(\frac{x}{7}+\frac{4}{15}=\frac{1}{4}x-3\)
12. \(\frac{x}{6}-\frac{5}{11}=\frac{-7}{5}x+6\)

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{2}{3}x-7 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{24}{ \color{blue}{36} }x-\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-20& = & 24x-252 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{-24x} & = & \color{red}{24x} -252 \color{blue}{-24x} \color{blue}{+20} \\\Leftrightarrow & -15x& = & -232 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-232}{-15} \\\Leftrightarrow & x = \frac{232}{15} & & \\ & V = \left\{ \frac{232}{15} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{7}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & 294x+630 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{-294x} & = & \color{red}{294x} +630 \color{blue}{-294x} \color{blue}{+75} \\\Leftrightarrow & -259x& = & 705 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{705}{-259} \\\Leftrightarrow & x = \frac{-705}{259} & & \\ & V = \left\{ \frac{-705}{259} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{3}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 126x+630 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-126x} & = & \color{red}{126x} +630 \color{blue}{-126x} \color{blue}{-120} \\\Leftrightarrow & -91x& = & 510 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{510}{-91} \\\Leftrightarrow & x = \frac{-510}{91} & & \\ & V = \left\{ \frac{-510}{91} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x+\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & -30x+18 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{+30x} & = & \color{red}{-30x} +18 \color{blue}{+30x} \color{blue}{-10} \\\Leftrightarrow & 39x& = & 8 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{8}{39} \\\Leftrightarrow & x = \frac{8}{39} & & \\ & V = \left\{ \frac{8}{39} \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-40}{ \color{blue}{24} }x-\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+15& = & -40x-144 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+40x} & = & \color{red}{-40x} -144 \color{blue}{+40x} \color{blue}{-15} \\\Leftrightarrow & 52x& = & -159 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{-159}{52} \\\Leftrightarrow & x = \frac{-159}{52} & & \\ & V = \left\{ \frac{-159}{52} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 10 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+10& = & 7x+56 \\\Leftrightarrow & 2x \color{red}{+10} \color{blue}{-10} \color{blue}{-7x} & = & \color{red}{7x} +56 \color{blue}{-7x} \color{blue}{-10} \\\Leftrightarrow & -5x& = & 46 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{46}{-5} \\\Leftrightarrow & x = \frac{-46}{5} & & \\ & V = \left\{ \frac{-46}{5} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -84x-1470 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+84x} & = & \color{red}{-84x} -1470 \color{blue}{+84x} \color{blue}{-90} \\\Leftrightarrow & 119x& = & -1560 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1560}{119} \\\Leftrightarrow & x = \frac{-1560}{119} & & \\ & V = \left\{ \frac{-1560}{119} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 42x+180 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-42x} & = & \color{red}{42x} +180 \color{blue}{-42x} \color{blue}{-8} \\\Leftrightarrow & -27x& = & 172 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{172}{-27} \\\Leftrightarrow & x = \frac{-172}{27} & & \\ & V = \left\{ \frac{-172}{27} \right\} & \\\end{align}\)
9. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & -220x-792 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{+220x} & = & \color{red}{-220x} -792 \color{blue}{+220x} \color{blue}{-60} \\\Leftrightarrow & 253x& = & -852 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{-852}{253} \\\Leftrightarrow & x = \frac{-852}{253} & & \\ & V = \left\{ \frac{-852}{253} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -126x-630 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+126x} & = & \color{red}{-126x} -630 \color{blue}{+126x} \color{blue}{+40} \\\Leftrightarrow & 141x& = & -590 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-590}{141} \\\Leftrightarrow & x = \frac{-590}{141} & & \\ & V = \left\{ \frac{-590}{141} \right\} & \\\end{align}\)
11. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 112 }{ \color{blue}{420} })& = & (\frac{105}{ \color{blue}{420} }x-\frac{1260}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+112& = & 105x-1260 \\\Leftrightarrow & 60x \color{red}{+112} \color{blue}{-112} \color{blue}{-105x} & = & \color{red}{105x} -1260 \color{blue}{-105x} \color{blue}{-112} \\\Leftrightarrow & -45x& = & -1372 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-1372}{-45} \\\Leftrightarrow & x = \frac{1372}{45} & & \\ & V = \left\{ \frac{1372}{45} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -462x+1980 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+462x} & = & \color{red}{-462x} +1980 \color{blue}{+462x} \color{blue}{+150} \\\Leftrightarrow & 517x& = & 2130 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{2130}{517} \\\Leftrightarrow & x = \frac{2130}{517} & & \\ & V = \left\{ \frac{2130}{517} \right\} & \\\end{align}\)