Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{2}{5}x+6 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }- \frac{ 70 }{ \color{blue}{455} })& = & (\frac{182}{ \color{blue}{455} }x+\frac{2730}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x-70& = & 182x+2730 \\\Leftrightarrow & 65x \color{red}{-70} \color{blue}{+70} \color{blue}{-182x} & = & \color{red}{182x} +2730 \color{blue}{-182x} \color{blue}{+70} \\\Leftrightarrow & -117x& = & 2800 \\\Leftrightarrow & \frac{-117x}{ \color{red}{-117} }& = & \frac{2800}{-117} \\\Leftrightarrow & x = \frac{-2800}{117} & & \\ & V = \left\{ \frac{-2800}{117} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 56x-336 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-56x} & = & \color{red}{56x} -336 \color{blue}{-56x} \color{blue}{+18} \\\Leftrightarrow & -35x& = & -318 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-318}{-35} \\\Leftrightarrow & x = \frac{318}{35} & & \\ & V = \left\{ \frac{318}{35} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-60& = & 39x+624 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{-39x} & = & \color{red}{39x} +624 \color{blue}{-39x} \color{blue}{+60} \\\Leftrightarrow & 13x& = & 684 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{684}{13} \\\Leftrightarrow & x = \frac{684}{13} & & \\ & V = \left\{ \frac{684}{13} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{-2}{5}x+7 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x+\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & -132x+2310 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{+132x} & = & \color{red}{-132x} +2310 \color{blue}{+132x} \color{blue}{-90} \\\Leftrightarrow & 187x& = & 2220 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{2220}{187} \\\Leftrightarrow & x = \frac{2220}{187} & & \\ & V = \left\{ \frac{2220}{187} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x-1680 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} -1680 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & -1560 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-1560}{-217} \\\Leftrightarrow & x = \frac{1560}{217} & & \\ & V = \left\{ \frac{1560}{217} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x+\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & -24x+72 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{+24x} & = & \color{red}{-24x} +72 \color{blue}{+24x} \color{blue}{+16} \\\Leftrightarrow & 33x& = & 88 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{88}{33} \\\Leftrightarrow & x = \frac{8}{3} & & \\ & V = \left\{ \frac{8}{3} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 63 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-63& = & -175x-1050 \\\Leftrightarrow & 30x \color{red}{-63} \color{blue}{+63} \color{blue}{+175x} & = & \color{red}{-175x} -1050 \color{blue}{+175x} \color{blue}{+63} \\\Leftrightarrow & 205x& = & -987 \\\Leftrightarrow & \frac{205x}{ \color{red}{205} }& = & \frac{-987}{205} \\\Leftrightarrow & x = \frac{-987}{205} & & \\ & V = \left\{ \frac{-987}{205} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -168x-630 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+168x} & = & \color{red}{-168x} -630 \color{blue}{+168x} \color{blue}{-90} \\\Leftrightarrow & 203x& = & -720 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-720}{203} \\\Leftrightarrow & x = \frac{-720}{203} & & \\ & V = \left\{ \frac{-720}{203} \right\} & \\\end{align}\)
9. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 105 }{ \color{blue}{231} })& = & (\frac{-385}{ \color{blue}{231} }x+\frac{693}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-105& = & -385x+693 \\\Leftrightarrow & 33x \color{red}{-105} \color{blue}{+105} \color{blue}{+385x} & = & \color{red}{-385x} +693 \color{blue}{+385x} \color{blue}{+105} \\\Leftrightarrow & 418x& = & 798 \\\Leftrightarrow & \frac{418x}{ \color{red}{418} }& = & \frac{798}{418} \\\Leftrightarrow & x = \frac{21}{11} & & \\ & V = \left\{ \frac{21}{11} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & -312x-1170 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+312x} & = & \color{red}{-312x} -1170 \color{blue}{+312x} \color{blue}{-60} \\\Leftrightarrow & 377x& = & -1230 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-1230}{377} \\\Leftrightarrow & x = \frac{-1230}{377} & & \\ & V = \left\{ \frac{-1230}{377} \right\} & \\\end{align}\)
11. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{3}{2}x+4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 21x+56 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-21x} & = & \color{red}{21x} +56 \color{blue}{-21x} \color{blue}{+4} \\\Leftrightarrow & -19x& = & 60 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{60}{-19} \\\Leftrightarrow & x = \frac{-60}{19} & & \\ & V = \left\{ \frac{-60}{19} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 468x+3120 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-468x} & = & \color{red}{468x} +3120 \color{blue}{-468x} \color{blue}{+120} \\\Leftrightarrow & -403x& = & 3240 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{3240}{-403} \\\Leftrightarrow & x = \frac{-3240}{403} & & \\ & V = \left\{ \frac{-3240}{403} \right\} & \\\end{align}\)