Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-30& = & 7x+126 \\\Leftrightarrow & 6x \color{red}{-30} \color{blue}{+30} \color{blue}{-7x} & = & \color{red}{7x} +126 \color{blue}{-7x} \color{blue}{+30} \\\Leftrightarrow & -x& = & 156 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{156}{-1} \\\Leftrightarrow & x = -156 & & \\ & V = \left\{ -156 \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{7}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 294x+1050 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-294x} & = & \color{red}{294x} +1050 \color{blue}{-294x} \color{blue}{-90} \\\Leftrightarrow & -259x& = & 960 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{960}{-259} \\\Leftrightarrow & x = \frac{-960}{259} & & \\ & V = \left\{ \frac{-960}{259} \right\} & \\\end{align}\)
3. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 30 }{ \color{blue}{110} })& = & (\frac{275}{ \color{blue}{110} }x+\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+30& = & 275x+770 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-275x} & = & \color{red}{275x} +770 \color{blue}{-275x} \color{blue}{-30} \\\Leftrightarrow & -253x& = & 740 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{740}{-253} \\\Leftrightarrow & x = \frac{-740}{253} & & \\ & V = \left\{ \frac{-740}{253} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{315}{ \color{blue}{90} }x-\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & 315x-270 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{-315x} & = & \color{red}{315x} -270 \color{blue}{-315x} \color{blue}{-20} \\\Leftrightarrow & -297x& = & -290 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{-290}{-297} \\\Leftrightarrow & x = \frac{290}{297} & & \\ & V = \left\{ \frac{290}{297} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x+\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & -64x+320 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{+64x} & = & \color{red}{-64x} +320 \color{blue}{+64x} \color{blue}{-15} \\\Leftrightarrow & 84x& = & 305 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{305}{84} \\\Leftrightarrow & x = \frac{305}{84} & & \\ & V = \left\{ \frac{305}{84} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 7, 6 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{7}{4}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 70 }{ \color{blue}{84} })& = & (\frac{147}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+70& = & 147x-588 \\\Leftrightarrow & 12x \color{red}{+70} \color{blue}{-70} \color{blue}{-147x} & = & \color{red}{147x} -588 \color{blue}{-147x} \color{blue}{-70} \\\Leftrightarrow & -135x& = & -658 \\\Leftrightarrow & \frac{-135x}{ \color{red}{-135} }& = & \frac{-658}{-135} \\\Leftrightarrow & x = \frac{658}{135} & & \\ & V = \left\{ \frac{658}{135} \right\} & \\\end{align}\)
7. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{7}{6}x-2 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 126 }{ \color{blue}{546} })& = & (\frac{637}{ \color{blue}{546} }x-\frac{1092}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-126& = & 637x-1092 \\\Leftrightarrow & 78x \color{red}{-126} \color{blue}{+126} \color{blue}{-637x} & = & \color{red}{637x} -1092 \color{blue}{-637x} \color{blue}{+126} \\\Leftrightarrow & -559x& = & -966 \\\Leftrightarrow & \frac{-559x}{ \color{red}{-559} }& = & \frac{-966}{-559} \\\Leftrightarrow & x = \frac{966}{559} & & \\ & V = \left\{ \frac{966}{559} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x-420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} -420 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & -360 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-360}{203} \\\Leftrightarrow & x = \frac{-360}{203} & & \\ & V = \left\{ \frac{-360}{203} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 98x+210 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-98x} & = & \color{red}{98x} +210 \color{blue}{-98x} \color{blue}{+12} \\\Leftrightarrow & -77x& = & 222 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{222}{-77} \\\Leftrightarrow & x = \frac{-222}{77} & & \\ & V = \left\{ \frac{-222}{77} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{2}{3}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{20}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 20x+210 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-20x} & = & \color{red}{20x} +210 \color{blue}{-20x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & 202 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{202}{-5} \\\Leftrightarrow & x = \frac{-202}{5} & & \\ & V = \left\{ \frac{-202}{5} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 52x+936 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-52x} & = & \color{red}{52x} +936 \color{blue}{-52x} \color{blue}{-48} \\\Leftrightarrow & -13x& = & 888 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{888}{-13} \\\Leftrightarrow & x = \frac{-888}{13} & & \\ & V = \left\{ \frac{-888}{13} \right\} & \\\end{align}\)
12. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 105 }{ \color{blue}{385} })& = & (\frac{77}{ \color{blue}{385} }x-\frac{2310}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-105& = & 77x-2310 \\\Leftrightarrow & 55x \color{red}{-105} \color{blue}{+105} \color{blue}{-77x} & = & \color{red}{77x} -2310 \color{blue}{-77x} \color{blue}{+105} \\\Leftrightarrow & -22x& = & -2205 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{-2205}{-22} \\\Leftrightarrow & x = \frac{2205}{22} & & \\ & V = \left\{ \frac{2205}{22} \right\} & \\\end{align}\)