Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{7}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 126x-90 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-126x} & = & \color{red}{126x} -90 \color{blue}{-126x} \color{blue}{-40} \\\Leftrightarrow & -111x& = & -130 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{-130}{-111} \\\Leftrightarrow & x = \frac{130}{111} & & \\ & V = \left\{ \frac{130}{111} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{-96}{ \color{blue}{36} }x-\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-20& = & -96x-72 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{+96x} & = & \color{red}{-96x} -72 \color{blue}{+96x} \color{blue}{+20} \\\Leftrightarrow & 105x& = & -52 \\\Leftrightarrow & \frac{105x}{ \color{red}{105} }& = & \frac{-52}{105} \\\Leftrightarrow & x = \frac{-52}{105} & & \\ & V = \left\{ \frac{-52}{105} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -42x-180 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+42x} & = & \color{red}{-42x} -180 \color{blue}{+42x} \color{blue}{-9} \\\Leftrightarrow & 47x& = & -189 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-189}{47} \\\Leftrightarrow & x = \frac{-189}{47} & & \\ & V = \left\{ \frac{-189}{47} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & -168x-840 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{+168x} & = & \color{red}{-168x} -840 \color{blue}{+168x} \color{blue}{+120} \\\Leftrightarrow & 203x& = & -720 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-720}{203} \\\Leftrightarrow & x = \frac{-720}{203} & & \\ & V = \left\{ \frac{-720}{203} \right\} & \\\end{align}\)
5. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 100 }{ \color{blue}{260} })& = & (\frac{-208}{ \color{blue}{260} }x+\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-100& = & -208x+1300 \\\Leftrightarrow & 65x \color{red}{-100} \color{blue}{+100} \color{blue}{+208x} & = & \color{red}{-208x} +1300 \color{blue}{+208x} \color{blue}{+100} \\\Leftrightarrow & 273x& = & 1400 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{1400}{273} \\\Leftrightarrow & x = \frac{200}{39} & & \\ & V = \left\{ \frac{200}{39} \right\} & \\\end{align}\)
6. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 50 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x-\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-50& = & -52x-390 \\\Leftrightarrow & 65x \color{red}{-50} \color{blue}{+50} \color{blue}{+52x} & = & \color{red}{-52x} -390 \color{blue}{+52x} \color{blue}{+50} \\\Leftrightarrow & 117x& = & -340 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-340}{117} \\\Leftrightarrow & x = \frac{-340}{117} & & \\ & V = \left\{ \frac{-340}{117} \right\} & \\\end{align}\)
7. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{3}{5}x-3 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{66}{ \color{blue}{110} }x-\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-20& = & 66x-330 \\\Leftrightarrow & 55x \color{red}{-20} \color{blue}{+20} \color{blue}{-66x} & = & \color{red}{66x} -330 \color{blue}{-66x} \color{blue}{+20} \\\Leftrightarrow & -11x& = & -310 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-310}{-11} \\\Leftrightarrow & x = \frac{310}{11} & & \\ & V = \left\{ \frac{310}{11} \right\} & \\\end{align}\)
8. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 84 }{ \color{blue}{273} })& = & (\frac{-455}{ \color{blue}{273} }x+\frac{1638}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+84& = & -455x+1638 \\\Leftrightarrow & 39x \color{red}{+84} \color{blue}{-84} \color{blue}{+455x} & = & \color{red}{-455x} +1638 \color{blue}{+455x} \color{blue}{-84} \\\Leftrightarrow & 494x& = & 1554 \\\Leftrightarrow & \frac{494x}{ \color{red}{494} }& = & \frac{1554}{494} \\\Leftrightarrow & x = \frac{777}{247} & & \\ & V = \left\{ \frac{777}{247} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{48}{ \color{blue}{36} }x+\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & 48x+288 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-48x} & = & \color{red}{48x} +288 \color{blue}{-48x} \color{blue}{+8} \\\Leftrightarrow & -39x& = & 296 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{296}{-39} \\\Leftrightarrow & x = \frac{-296}{39} & & \\ & V = \left\{ \frac{-296}{39} \right\} & \\\end{align}\)
10. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{325}{ \color{blue}{195} }x-\frac{195}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+30& = & 325x-195 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{-325x} & = & \color{red}{325x} -195 \color{blue}{-325x} \color{blue}{-30} \\\Leftrightarrow & -286x& = & -225 \\\Leftrightarrow & \frac{-286x}{ \color{red}{-286} }& = & \frac{-225}{-286} \\\Leftrightarrow & x = \frac{225}{286} & & \\ & V = \left\{ \frac{225}{286} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 126x+540 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-126x} & = & \color{red}{126x} +540 \color{blue}{-126x} \color{blue}{-20} \\\Leftrightarrow & -111x& = & 520 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{520}{-111} \\\Leftrightarrow & x = \frac{-520}{111} & & \\ & V = \left\{ \frac{-520}{111} \right\} & \\\end{align}\)
12. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{24}{ \color{blue}{18} }x+\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-10& = & 24x+90 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-24x} & = & \color{red}{24x} +90 \color{blue}{-24x} \color{blue}{+10} \\\Leftrightarrow & -15x& = & 100 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{100}{-15} \\\Leftrightarrow & x = \frac{-20}{3} & & \\ & V = \left\{ \frac{-20}{3} \right\} & \\\end{align}\)