Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{5}{3}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{260}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & 260x-312 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-260x} & = & \color{red}{260x} -312 \color{blue}{-260x} \color{blue}{+24} \\\Leftrightarrow & -221x& = & -288 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{-288}{-221} \\\Leftrightarrow & x = \frac{288}{221} & & \\ & V = \left\{ \frac{288}{221} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 3 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x-\frac{42}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-3& = & 21x-42 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-21x} & = & \color{red}{21x} -42 \color{blue}{-21x} \color{blue}{+3} \\\Leftrightarrow & -19x& = & -39 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-39}{-19} \\\Leftrightarrow & x = \frac{39}{19} & & \\ & V = \left\{ \frac{39}{19} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-21}{ \color{blue}{12} }x+\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & -21x+84 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{+21x} & = & \color{red}{-21x} +84 \color{blue}{+21x} \color{blue}{-5} \\\Leftrightarrow & 25x& = & 79 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{79}{25} \\\Leftrightarrow & x = \frac{79}{25} & & \\ & V = \left\{ \frac{79}{25} \right\} & \\\end{align}\)
4. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{20}{ \color{blue}{120} }x-\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-75& = & 20x-480 \\\Leftrightarrow & 24x \color{red}{-75} \color{blue}{+75} \color{blue}{-20x} & = & \color{red}{20x} -480 \color{blue}{-20x} \color{blue}{+75} \\\Leftrightarrow & 4x& = & -405 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-405}{4} \\\Leftrightarrow & x = \frac{-405}{4} & & \\ & V = \left\{ \frac{-405}{4} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & -168x+1470 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{+168x} & = & \color{red}{-168x} +1470 \color{blue}{+168x} \color{blue}{+120} \\\Leftrightarrow & 203x& = & 1590 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1590}{203} \\\Leftrightarrow & x = \frac{1590}{203} & & \\ & V = \left\{ \frac{1590}{203} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x+210 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} +210 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 234 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{234}{-7} \\\Leftrightarrow & x = \frac{-234}{7} & & \\ & V = \left\{ \frac{-234}{7} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{9}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+50& = & 108x+540 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{-108x} & = & \color{red}{108x} +540 \color{blue}{-108x} \color{blue}{-50} \\\Leftrightarrow & -93x& = & 490 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{490}{-93} \\\Leftrightarrow & x = \frac{-490}{93} & & \\ & V = \left\{ \frac{-490}{93} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-56}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & -56x+700 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+56x} & = & \color{red}{-56x} +700 \color{blue}{+56x} \color{blue}{-60} \\\Leftrightarrow & 91x& = & 640 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{640}{91} \\\Leftrightarrow & x = \frac{640}{91} & & \\ & V = \left\{ \frac{640}{91} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+18& = & 39x+312 \\\Leftrightarrow & 26x \color{red}{+18} \color{blue}{-18} \color{blue}{-39x} & = & \color{red}{39x} +312 \color{blue}{-39x} \color{blue}{-18} \\\Leftrightarrow & -13x& = & 294 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{294}{-13} \\\Leftrightarrow & x = \frac{-294}{13} & & \\ & V = \left\{ \frac{-294}{13} \right\} & \\\end{align}\)
10. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+20& = & 55x-220 \\\Leftrightarrow & 22x \color{red}{+20} \color{blue}{-20} \color{blue}{-55x} & = & \color{red}{55x} -220 \color{blue}{-55x} \color{blue}{-20} \\\Leftrightarrow & -33x& = & -240 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-240}{-33} \\\Leftrightarrow & x = \frac{80}{11} & & \\ & V = \left\{ \frac{80}{11} \right\} & \\\end{align}\)
11. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{455}{ \color{blue}{130} }x+\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-30& = & 455x+390 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-455x} & = & \color{red}{455x} +390 \color{blue}{-455x} \color{blue}{+30} \\\Leftrightarrow & -429x& = & 420 \\\Leftrightarrow & \frac{-429x}{ \color{red}{-429} }& = & \frac{420}{-429} \\\Leftrightarrow & x = \frac{-140}{143} & & \\ & V = \left\{ \frac{-140}{143} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 100 }{ \color{blue}{140} })& = & (\frac{196}{ \color{blue}{140} }x+\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+100& = & 196x+980 \\\Leftrightarrow & 35x \color{red}{+100} \color{blue}{-100} \color{blue}{-196x} & = & \color{red}{196x} +980 \color{blue}{-196x} \color{blue}{-100} \\\Leftrightarrow & -161x& = & 880 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{880}{-161} \\\Leftrightarrow & x = \frac{-880}{161} & & \\ & V = \left\{ \frac{-880}{161} \right\} & \\\end{align}\)