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

---

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & -264x-2640 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{+264x} & = & \color{red}{-264x} -2640 \color{blue}{+264x} \color{blue}{-120} \\\Leftrightarrow & 319x& = & -2760 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-2760}{319} \\\Leftrightarrow & x = \frac{-2760}{319} & & \\ & V = \left\{ \frac{-2760}{319} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{7}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 294x-630 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-294x} & = & \color{red}{294x} -630 \color{blue}{-294x} \color{blue}{-90} \\\Leftrightarrow & -259x& = & -720 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-720}{-259} \\\Leftrightarrow & x = \frac{720}{259} & & \\ & V = \left\{ \frac{720}{259} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x+\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & -64x+480 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{+64x} & = & \color{red}{-64x} +480 \color{blue}{+64x} \color{blue}{-15} \\\Leftrightarrow & 84x& = & 465 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{465}{84} \\\Leftrightarrow & x = \frac{155}{28} & & \\ & V = \left\{ \frac{155}{28} \right\} & \\\end{align}\)
4. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{-520}{ \color{blue}{195} }x-\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+60& = & -520x-390 \\\Leftrightarrow & 39x \color{red}{+60} \color{blue}{-60} \color{blue}{+520x} & = & \color{red}{-520x} -390 \color{blue}{+520x} \color{blue}{-60} \\\Leftrightarrow & 559x& = & -450 \\\Leftrightarrow & \frac{559x}{ \color{red}{559} }& = & \frac{-450}{559} \\\Leftrightarrow & x = \frac{-450}{559} & & \\ & V = \left\{ \frac{-450}{559} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{-42}{ \color{blue}{105} }x-\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-45& = & -42x-105 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+42x} & = & \color{red}{-42x} -105 \color{blue}{+42x} \color{blue}{+45} \\\Leftrightarrow & 77x& = & -60 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-60}{77} \\\Leftrightarrow & x = \frac{-60}{77} & & \\ & V = \left\{ \frac{-60}{77} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x+420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} +420 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & 480 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{480}{203} \\\Leftrightarrow & x = \frac{480}{203} & & \\ & V = \left\{ \frac{480}{203} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{2}{3}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 56x-420 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-56x} & = & \color{red}{56x} -420 \color{blue}{-56x} \color{blue}{-48} \\\Leftrightarrow & -35x& = & -468 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-468}{-35} \\\Leftrightarrow & x = \frac{468}{35} & & \\ & V = \left\{ \frac{468}{35} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x+156 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} +156 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & 144 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{144}{-13} \\\Leftrightarrow & x = \frac{-144}{13} & & \\ & V = \left\{ \frac{-144}{13} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 50 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x-\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+50& = & -98x-70 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{+98x} & = & \color{red}{-98x} -70 \color{blue}{+98x} \color{blue}{-50} \\\Leftrightarrow & 133x& = & -120 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-120}{133} \\\Leftrightarrow & x = \frac{-120}{133} & & \\ & V = \left\{ \frac{-120}{133} \right\} & \\\end{align}\)
10. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{14}{ \color{blue}{6} }x+\frac{36}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 14x+36 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-14x} & = & \color{red}{14x} +36 \color{blue}{-14x} \color{blue}{-5} \\\Leftrightarrow & -11x& = & 31 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{31}{-11} \\\Leftrightarrow & x = \frac{-31}{11} & & \\ & V = \left\{ \frac{-31}{11} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{5}{6}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+12& = & 35x+294 \\\Leftrightarrow & 6x \color{red}{+12} \color{blue}{-12} \color{blue}{-35x} & = & \color{red}{35x} +294 \color{blue}{-35x} \color{blue}{-12} \\\Leftrightarrow & -29x& = & 282 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{282}{-29} \\\Leftrightarrow & x = \frac{-282}{29} & & \\ & V = \left\{ \frac{-282}{29} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-8& = & 15x+180 \\\Leftrightarrow & 20x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} +180 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & 5x& = & 188 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{188}{5} \\\Leftrightarrow & x = \frac{188}{5} & & \\ & V = \left\{ \frac{188}{5} \right\} & \\\end{align}\)