Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-60& = & -175x+210 \\\Leftrightarrow & 21x \color{red}{-60} \color{blue}{+60} \color{blue}{+175x} & = & \color{red}{-175x} +210 \color{blue}{+175x} \color{blue}{+60} \\\Leftrightarrow & 196x& = & 270 \\\Leftrightarrow & \frac{196x}{ \color{red}{196} }& = & \frac{270}{196} \\\Leftrightarrow & x = \frac{135}{98} & & \\ & V = \left\{ \frac{135}{98} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 35x+140 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-35x} & = & \color{red}{35x} +140 \color{blue}{-35x} \color{blue}{-30} \\\Leftrightarrow & -21x& = & 110 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{110}{-21} \\\Leftrightarrow & x = \frac{-110}{21} & & \\ & V = \left\{ \frac{-110}{21} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{48}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & 48x+560 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{-48x} & = & \color{red}{48x} +560 \color{blue}{-48x} \color{blue}{+25} \\\Leftrightarrow & -8x& = & 585 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{585}{-8} \\\Leftrightarrow & x = \frac{-585}{8} & & \\ & V = \left\{ \frac{-585}{8} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -312x-780 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+312x} & = & \color{red}{-312x} -780 \color{blue}{+312x} \color{blue}{-90} \\\Leftrightarrow & 377x& = & -870 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-870}{377} \\\Leftrightarrow & x = \frac{-30}{13} & & \\ & V = \left\{ \frac{-30}{13} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 126x-1260 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-126x} & = & \color{red}{126x} -1260 \color{blue}{-126x} \color{blue}{-60} \\\Leftrightarrow & -91x& = & -1320 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-1320}{-91} \\\Leftrightarrow & x = \frac{1320}{91} & & \\ & V = \left\{ \frac{1320}{91} \right\} & \\\end{align}\)
6. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+14& = & -70x-735 \\\Leftrightarrow & 15x \color{red}{+14} \color{blue}{-14} \color{blue}{+70x} & = & \color{red}{-70x} -735 \color{blue}{+70x} \color{blue}{-14} \\\Leftrightarrow & 85x& = & -749 \\\Leftrightarrow & \frac{85x}{ \color{red}{85} }& = & \frac{-749}{85} \\\Leftrightarrow & x = \frac{-749}{85} & & \\ & V = \left\{ \frac{-749}{85} \right\} & \\\end{align}\)
7. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{5}{4}x-3 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{325}{ \color{blue}{260} }x-\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & 325x-780 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{-325x} & = & \color{red}{325x} -780 \color{blue}{-325x} \color{blue}{+60} \\\Leftrightarrow & -273x& = & -720 \\\Leftrightarrow & \frac{-273x}{ \color{red}{-273} }& = & \frac{-720}{-273} \\\Leftrightarrow & x = \frac{240}{91} & & \\ & V = \left\{ \frac{240}{91} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{264}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & 264x-1650 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{-264x} & = & \color{red}{264x} -1650 \color{blue}{-264x} \color{blue}{-90} \\\Leftrightarrow & -209x& = & -1740 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-1740}{-209} \\\Leftrightarrow & x = \frac{1740}{209} & & \\ & V = \left\{ \frac{1740}{209} \right\} & \\\end{align}\)
9. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{-7}{4}x+3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 12 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x+\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+12& = & -49x+84 \\\Leftrightarrow & 4x \color{red}{+12} \color{blue}{-12} \color{blue}{+49x} & = & \color{red}{-49x} +84 \color{blue}{+49x} \color{blue}{-12} \\\Leftrightarrow & 53x& = & 72 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{72}{53} \\\Leftrightarrow & x = \frac{72}{53} & & \\ & V = \left\{ \frac{72}{53} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 140x+588 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-140x} & = & \color{red}{140x} +588 \color{blue}{-140x} \color{blue}{+48} \\\Leftrightarrow & -119x& = & 636 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{636}{-119} \\\Leftrightarrow & x = \frac{-636}{119} & & \\ & V = \left\{ \frac{-636}{119} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x-\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & -220x-528 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{+220x} & = & \color{red}{-220x} -528 \color{blue}{+220x} \color{blue}{-60} \\\Leftrightarrow & 253x& = & -588 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{-588}{253} \\\Leftrightarrow & x = \frac{-588}{253} & & \\ & V = \left\{ \frac{-588}{253} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & 26x-546 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-26x} & = & \color{red}{26x} -546 \color{blue}{-26x} \color{blue}{-24} \\\Leftrightarrow & 13x& = & -570 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-570}{13} \\\Leftrightarrow & x = \frac{-570}{13} & & \\ & V = \left\{ \frac{-570}{13} \right\} & \\\end{align}\)