Vgln. eerste graad (reeks 4)

Hoofdmenu Eentje per keer  🖨

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

1. \(\frac{x}{6}+\frac{3}{7}=\frac{-7}{5}x+4\)
2. \(\frac{x}{4}+\frac{4}{7}=\frac{1}{5}x-1\)
3. \(\frac{x}{2}-\frac{4}{9}=\frac{-5}{3}x-1\)
4. \(\frac{x}{4}-\frac{3}{7}=\frac{2}{3}x-3\)
5. \(\frac{x}{6}+\frac{5}{14}=\frac{4}{5}x+4\)
6. \(\frac{x}{3}-\frac{5}{7}=\frac{-7}{4}x-5\)
7. \(\frac{x}{2}+\frac{4}{13}=\frac{1}{3}x-6\)
8. \(\frac{x}{6}+\frac{5}{6}=\frac{-2}{5}x+5\)
9. \(\frac{x}{5}-\frac{5}{8}=\frac{1}{4}x+7\)
10. \(\frac{x}{5}+\frac{4}{11}=\frac{-5}{6}x+8\)
11. \(\frac{x}{3}-\frac{5}{6}=\frac{6}{5}x+6\)
12. \(\frac{x}{3}-\frac{2}{15}=\frac{3}{4}x+1\)

---

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -294x+840 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+294x} & = & \color{red}{-294x} +840 \color{blue}{+294x} \color{blue}{-90} \\\Leftrightarrow & 329x& = & 750 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{750}{329} \\\Leftrightarrow & x = \frac{750}{329} & & \\ & V = \left\{ \frac{750}{329} \right\} & \\\end{align}\)
2. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 28x-140 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-28x} & = & \color{red}{28x} -140 \color{blue}{-28x} \color{blue}{-80} \\\Leftrightarrow & 7x& = & -220 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-220}{7} \\\Leftrightarrow & x = \frac{-220}{7} & & \\ & V = \left\{ \frac{-220}{7} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x-\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -30x-18 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+30x} & = & \color{red}{-30x} -18 \color{blue}{+30x} \color{blue}{+8} \\\Leftrightarrow & 39x& = & -10 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-10}{39} \\\Leftrightarrow & x = \frac{-10}{39} & & \\ & V = \left\{ \frac{-10}{39} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{2}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 56x-252 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-56x} & = & \color{red}{56x} -252 \color{blue}{-56x} \color{blue}{+36} \\\Leftrightarrow & -35x& = & -216 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-216}{-35} \\\Leftrightarrow & x = \frac{216}{35} & & \\ & V = \left\{ \frac{216}{35} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{4}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & 168x+840 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{-168x} & = & \color{red}{168x} +840 \color{blue}{-168x} \color{blue}{-75} \\\Leftrightarrow & -133x& = & 765 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{765}{-133} \\\Leftrightarrow & x = \frac{-765}{133} & & \\ & V = \left\{ \frac{-765}{133} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{7}& = & \frac{-7}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 60 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-60& = & -147x-420 \\\Leftrightarrow & 28x \color{red}{-60} \color{blue}{+60} \color{blue}{+147x} & = & \color{red}{-147x} -420 \color{blue}{+147x} \color{blue}{+60} \\\Leftrightarrow & 175x& = & -360 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-360}{175} \\\Leftrightarrow & x = \frac{-72}{35} & & \\ & V = \left\{ \frac{-72}{35} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & 26x-468 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-26x} & = & \color{red}{26x} -468 \color{blue}{-26x} \color{blue}{-24} \\\Leftrightarrow & 13x& = & -492 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-492}{13} \\\Leftrightarrow & x = \frac{-492}{13} & & \\ & V = \left\{ \frac{-492}{13} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -12x+150 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+12x} & = & \color{red}{-12x} +150 \color{blue}{+12x} \color{blue}{-25} \\\Leftrightarrow & 17x& = & 125 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{125}{17} \\\Leftrightarrow & x = \frac{125}{17} & & \\ & V = \left\{ \frac{125}{17} \right\} & \\\end{align}\)
9. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{1}{4}x+7 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{10}{ \color{blue}{40} }x+\frac{280}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x-25& = & 10x+280 \\\Leftrightarrow & 8x \color{red}{-25} \color{blue}{+25} \color{blue}{-10x} & = & \color{red}{10x} +280 \color{blue}{-10x} \color{blue}{+25} \\\Leftrightarrow & -2x& = & 305 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{305}{-2} \\\Leftrightarrow & x = \frac{-305}{2} & & \\ & V = \left\{ \frac{-305}{2} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{-5}{6}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & -275x+2640 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{+275x} & = & \color{red}{-275x} +2640 \color{blue}{+275x} \color{blue}{-120} \\\Leftrightarrow & 341x& = & 2520 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{2520}{341} \\\Leftrightarrow & x = \frac{2520}{341} & & \\ & V = \left\{ \frac{2520}{341} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 3, 6 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-25& = & 36x+180 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-36x} & = & \color{red}{36x} +180 \color{blue}{-36x} \color{blue}{+25} \\\Leftrightarrow & -26x& = & 205 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{205}{-26} \\\Leftrightarrow & x = \frac{-205}{26} & & \\ & V = \left\{ \frac{-205}{26} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{3}{4}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{45}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-8& = & 45x+60 \\\Leftrightarrow & 20x \color{red}{-8} \color{blue}{+8} \color{blue}{-45x} & = & \color{red}{45x} +60 \color{blue}{-45x} \color{blue}{+8} \\\Leftrightarrow & -25x& = & 68 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{68}{-25} \\\Leftrightarrow & x = \frac{-68}{25} & & \\ & V = \left\{ \frac{-68}{25} \right\} & \\\end{align}\)