Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & 21x+336 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{-21x} & = & \color{red}{21x} +336 \color{blue}{-21x} \color{blue}{-48} \\\Leftrightarrow & 7x& = & 288 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{288}{7} \\\Leftrightarrow & x = \frac{288}{7} & & \\ & V = \left\{ \frac{288}{7} \right\} & \\\end{align}\)
2. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{-5}{6}x+5 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 70 }{ \color{blue}{126} })& = & (\frac{-105}{ \color{blue}{126} }x+\frac{630}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+70& = & -105x+630 \\\Leftrightarrow & 18x \color{red}{+70} \color{blue}{-70} \color{blue}{+105x} & = & \color{red}{-105x} +630 \color{blue}{+105x} \color{blue}{-70} \\\Leftrightarrow & 123x& = & 560 \\\Leftrightarrow & \frac{123x}{ \color{red}{123} }& = & \frac{560}{123} \\\Leftrightarrow & x = \frac{560}{123} & & \\ & V = \left\{ \frac{560}{123} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -28x-42 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+28x} & = & \color{red}{-28x} -42 \color{blue}{+28x} \color{blue}{-24} \\\Leftrightarrow & 49x& = & -66 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-66}{49} \\\Leftrightarrow & x = \frac{-66}{49} & & \\ & V = \left\{ \frac{-66}{49} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{165}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+20& = & 165x+660 \\\Leftrightarrow & 22x \color{red}{+20} \color{blue}{-20} \color{blue}{-165x} & = & \color{red}{165x} +660 \color{blue}{-165x} \color{blue}{-20} \\\Leftrightarrow & -143x& = & 640 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{640}{-143} \\\Leftrightarrow & x = \frac{-640}{143} & & \\ & V = \left\{ \frac{-640}{143} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{4}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 168x+1050 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-168x} & = & \color{red}{168x} +1050 \color{blue}{-168x} \color{blue}{-90} \\\Leftrightarrow & -133x& = & 960 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{960}{-133} \\\Leftrightarrow & x = \frac{-960}{133} & & \\ & V = \left\{ \frac{-960}{133} \right\} & \\\end{align}\)
6. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 60 }{ \color{blue}{165} })& = & (\frac{-275}{ \color{blue}{165} }x-\frac{1155}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-60& = & -275x-1155 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{+275x} & = & \color{red}{-275x} -1155 \color{blue}{+275x} \color{blue}{+60} \\\Leftrightarrow & 308x& = & -1095 \\\Leftrightarrow & \frac{308x}{ \color{red}{308} }& = & \frac{-1095}{308} \\\Leftrightarrow & x = \frac{-1095}{308} & & \\ & V = \left\{ \frac{-1095}{308} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{4}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{56}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & 56x+70 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{-56x} & = & \color{red}{56x} +70 \color{blue}{-56x} \color{blue}{+30} \\\Leftrightarrow & -21x& = & 100 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{100}{-21} \\\Leftrightarrow & x = \frac{-100}{21} & & \\ & V = \left\{ \frac{-100}{21} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 80x+120 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-80x} & = & \color{red}{80x} +120 \color{blue}{-80x} \color{blue}{+8} \\\Leftrightarrow & -65x& = & 128 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{128}{-65} \\\Leftrightarrow & x = \frac{-128}{65} & & \\ & V = \left\{ \frac{-128}{65} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{-5}{6}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+90& = & -325x+1560 \\\Leftrightarrow & 78x \color{red}{+90} \color{blue}{-90} \color{blue}{+325x} & = & \color{red}{-325x} +1560 \color{blue}{+325x} \color{blue}{-90} \\\Leftrightarrow & 403x& = & 1470 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{1470}{403} \\\Leftrightarrow & x = \frac{1470}{403} & & \\ & V = \left\{ \frac{1470}{403} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{104}{ \color{blue}{130} }x+\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & 104x+910 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{-104x} & = & \color{red}{104x} +910 \color{blue}{-104x} \color{blue}{+30} \\\Leftrightarrow & -39x& = & 940 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{940}{-39} \\\Leftrightarrow & x = \frac{-940}{39} & & \\ & V = \left\{ \frac{-940}{39} \right\} & \\\end{align}\)
11. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }- \frac{ 140 }{ \color{blue}{308} })& = & (\frac{77}{ \color{blue}{308} }x-\frac{924}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x-140& = & 77x-924 \\\Leftrightarrow & 44x \color{red}{-140} \color{blue}{+140} \color{blue}{-77x} & = & \color{red}{77x} -924 \color{blue}{-77x} \color{blue}{+140} \\\Leftrightarrow & -33x& = & -784 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-784}{-33} \\\Leftrightarrow & x = \frac{784}{33} & & \\ & V = \left\{ \frac{784}{33} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{52}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & 52x+156 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{-52x} & = & \color{red}{52x} +156 \color{blue}{-52x} \color{blue}{+12} \\\Leftrightarrow & -13x& = & 168 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{168}{-13} \\\Leftrightarrow & x = \frac{-168}{13} & & \\ & V = \left\{ \frac{-168}{13} \right\} & \\\end{align}\)