Matemáticas 2 · Tema 5

Ecuaciones con fracciones

Video

Ejemplos

Operaciones Directas con Fracciones 3(12−45x)−5x=25x+332−125x−5x=25x+3−125x−5x−25x=3−32−145x−5x=6−32(−145−5)x=32−395x=32x=32(5−39)x=−526≈−0.19\begin{aligned} 3\left(\frac{1}{2} - \frac{4}{5}x\right) - 5x &= \frac{2}{5}x + 3 \\[14pt] \frac{3}{2} - \frac{12}{5}x - 5x &= \frac{2}{5}x + 3 \\[14pt] -\frac{12}{5}x - 5x - \frac{2}{5}x &= 3 - \frac{3}{2} \\[14pt] -\frac{14}{5}x - 5x &= \frac{6 - 3}{2} \\[14pt] \left(-\frac{14}{5} - 5\right)x &= \frac{3}{2} \\[14pt] -\frac{39}{5}x &= \frac{3}{2} \\[14pt] x &= \frac{3}{2} \left(\frac{5}{-39}\right) \\[14pt] \mathbf{x = -\frac{5}{26} \approx -0.19} \end{aligned}
Igualando denominador −6x+43x=75−183x+43x=75−143x=75−14(5)=7(3x)−70=21x−7021=xx=−103≈−3.33\begin{aligned} -\frac{6}{x} + \frac{4}{3x} &= \frac{7}{5} \\[14pt] -\frac{18}{3x} + \frac{4}{3x} &= \frac{7}{5} \\[14pt] \frac{-14}{3x} &= \frac{7}{5} \\[14pt] -14(5) &= 7(3x) \\[14pt] -70 &= 21x \\[14pt] \frac{-70}{21} &= x \\[14pt] \mathbf{x = -\frac{10}{3} \approx -3.33} \end{aligned}
Multiplicación por el MCM 42x−35=−14x−4360(42x−35)=60(−14x−43)2402x−1805=−604x−2403120x−36=−15x−80120x+15x=−80+36135x=−44x=−44135≈−0.33\begin{aligned} \frac{4}{2}x - \frac{3}{5} &= -\frac{1}{4}x - \frac{4}{3} \\[14pt] 60 \left(\frac{4}{2}x - \frac{3}{5}\right) &= 60 \left(-\frac{1}{4}x - \frac{4}{3}\right) \\[14pt] \frac{240}{2}x - \frac{180}{5} &= -\frac{60}{4}x - \frac{240}{3} \\[14pt] 120x - 36 &= -15x - 80 \\[14pt] 120x + 15x &= -80 + 36 \\[14pt] 135x &= -44 \\[14pt] \mathbf{x = -\frac{44}{135} \approx -0.33} \end{aligned}

Ejercicios

Bloque 1
−32x−46x=45-\dfrac{3}{2x} - \dfrac{4}{6x} = \dfrac{4}{5}
32x−25=−34x−65\dfrac{3}{2}x - \dfrac{2}{5} = -\dfrac{3}{4}x - \dfrac{6}{5}
2(56+38x)−25x=6x−22\left(\dfrac{5}{6} + \dfrac{3}{8}x\right) - \dfrac{2}{5}x = 6x - 2
Bloque 2
−54x+2x=−13-\dfrac{5}{4x} + \dfrac{2}{x} = -\dfrac{1}{3}
310x−15=4x3−12\dfrac{3}{10}x - \dfrac{1}{5} = \dfrac{4x}{3} - \dfrac{1}{2}
56x−4=73−2(13−54x)\dfrac{5}{6}x - 4 = \dfrac{7}{3} - 2\left(\dfrac{1}{3} - \dfrac{5}{4}x\right)
Bloque 3
−35x+24x=−110-\dfrac{3}{5x} + \dfrac{2}{4x} = -\dfrac{1}{10}
−23+45x=−23x+52-\dfrac{2}{3} + \dfrac{4}{5}x = -\dfrac{2}{3}x + \dfrac{5}{2}
−32x−4(53−3x)=−42x−65-\dfrac{3}{2}x - 4\left(\dfrac{5}{3} - 3x\right) = -\dfrac{4}{2}x - \dfrac{6}{5}
Bloque 4
−63x+24x=−12-\dfrac{6}{3x} + \dfrac{2}{4x} = -\dfrac{1}{2}
26−12x=53−34x\dfrac{2}{6} - \dfrac{1}{2}x = \dfrac{5}{3} - \dfrac{3}{4}x
34−52x=35−4(25x−12)\dfrac{3}{4} - \dfrac{5}{2}x = \dfrac{3}{5} - 4\left(\dfrac{2}{5}x - \dfrac{1}{2}\right)

Resoluciones

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Bloque 1
−32x−46x=45−96x−46x=45−136x=45−65=24xx=−6524≈−2.71\begin{aligned} -\frac{3}{2x} - \frac{4}{6x} &= \frac{4}{5} \\[10pt] -\frac{9}{6x} - \frac{4}{6x} &= \frac{4}{5} \\[10pt] \frac{-13}{6x} &= \frac{4}{5} \\[10pt] -65 &= 24x \\[10pt] \mathbf{x = -\frac{65}{24} \approx -2.71} \end{aligned}
32x−25=−34x−6520(32x−25)=20(−34x−65)30x−8=−15x−2445x=−16x=−1645≈−0.36\begin{aligned} \frac{3}{2}x - \frac{2}{5} &= -\frac{3}{4}x - \frac{6}{5} \\[10pt] 20\left(\frac{3}{2}x - \frac{2}{5}\right) &= 20\left(-\frac{3}{4}x - \frac{6}{5}\right) \\[10pt] 30x - 8 &= -15x - 24 \\[10pt] 45x &= -16 \\[10pt] \mathbf{x = -\frac{16}{45} \approx -0.36} \end{aligned}
2(56+38x)−25x=6x−253+34x−25x=6x−260(53+34x−25x)=60(6x−2)100+45x−24x=360x−120100+21x=360x−120220=339xx=220339≈0.65\begin{aligned} 2\left(\frac{5}{6} + \frac{3}{8}x\right) - \frac{2}{5}x &= 6x - 2 \\[10pt] \frac{5}{3} + \frac{3}{4}x - \frac{2}{5}x &= 6x - 2 \\[10pt] 60 \left(\frac{5}{3} + \frac{3}{4}x - \frac{2}{5}x\right) &= 60(6x - 2) \\[10pt] 100 + 45x - 24x &= 360x - 120 \\[10pt] 100 + 21x &= 360x - 120 \\[10pt] 220 &= 339x \\[10pt] \mathbf{x = \frac{220}{339} \approx 0.65} \end{aligned}
Bloque 2
−54x+2x=−13−54x+84x=−1334x=−139=−4xx=−94=−2.25\begin{aligned} -\frac{5}{4x} + \frac{2}{x} &= -\frac{1}{3} \\[10pt] -\frac{5}{4x} + \frac{8}{4x} &= -\frac{1}{3} \\[10pt] \frac{3}{4x} &= -\frac{1}{3} \\[10pt] 9 &= -4x \\[10pt] \mathbf{x = -\frac{9}{4} = -2.25} \end{aligned}
310x−15=4x3−1230(310x−15)=30(4x3−12)9x−6=40x−159=31x−159=31xx=931≈0.29\begin{aligned} \frac{3}{10}x - \frac{1}{5} &= \frac{4x}{3} - \frac{1}{2} \\[10pt] 30\left(\frac{3}{10}x - \frac{1}{5}\right) &= 30\left(\frac{4x}{3} - \frac{1}{2}\right) \\[10pt] 9x - 6 &= 40x - 15 \\[10pt] 9 &= 31x - 15 \\[10pt] 9 &= 31x \\[10pt] \mathbf{x = \frac{9}{31} \approx 0.29} \end{aligned}
56x−4=73−2(13−54x)56x−4=73−23+52x56x−4=53+52x6(56x−4)=6(53+52x)5x−24=10+15x−34=10xx=−3410x=−3.4\begin{aligned} \frac{5}{6}x - 4 &= \frac{7}{3} - 2\left(\frac{1}{3} - \frac{5}{4}x\right) \\[10pt] \frac{5}{6}x - 4 &= \frac{7}{3} - \frac{2}{3} + \frac{5}{2}x \\[10pt] \frac{5}{6}x - 4 &= \frac{5}{3} + \frac{5}{2}x \\[10pt] 6\left(\frac{5}{6}x - 4\right) &= 6\left(\frac{5}{3} + \frac{5}{2}x\right) \\[10pt] 5x - 24 &= 10 + 15x \\[10pt] -34 &= 10x \\[10pt] x &= -\frac{34}{10} \\[10pt] \mathbf{x = -3.4} \end{aligned}
Bloque 3
−35x+24x=−110−610x+510x=−110−110x=−110−10=−10xx=1\begin{aligned} -\frac{3}{5x} + \frac{2}{4x} &= -\frac{1}{10} \\[10pt] -\frac{6}{10x} + \frac{5}{10x} &= -\frac{1}{10} \\[10pt] \frac{-1}{10x} &= -\frac{1}{10} \\[10pt] -10 &= -10x \\[10pt] \mathbf{x = 1} \end{aligned}
−23+45x=−23x+5230(−23+45x)=30(−23x+52)−20+24x=−20x+7544x=95x=9544≈2.16\begin{aligned} -\frac{2}{3} + \frac{4}{5}x &= -\frac{2}{3}x + \frac{5}{2} \\[10pt] 30 \left(-\frac{2}{3} + \frac{4}{5}x\right) &= 30 \left(-\frac{2}{3}x + \frac{5}{2}\right) \\[10pt] -20 + 24x &= -20x + 75 \\[10pt] 44x &= 95 \\[10pt] \mathbf{x = \frac{95}{44} \approx 2.16} \end{aligned}
−32x−4(53−3x)=−42x−65−32x−203+12x=−2x−6530(−32x−203+12x)=30(−2x−65)−45x−200+360x=−60x−36315x−200=−60x−36375x=164x=164375≈0.44\begin{aligned} -\frac{3}{2}x - 4\left(\frac{5}{3} - 3x\right) &= -\frac{4}{2}x - \frac{6}{5} \\[10pt] -\frac{3}{2}x - \frac{20}{3} + 12x &= -2x - \frac{6}{5} \\[10pt] 30\left(-\frac{3}{2}x - \frac{20}{3} + 12x\right) &= 30\left(-2x - \frac{6}{5}\right) \\[10pt] -45x - 200 + 360x &= -60x - 36 \\[10pt] 315x - 200 &= -60x - 36 \\[10pt] 375x &= 164 \\[10pt] \mathbf{x = \frac{164}{375} \approx 0.44} \end{aligned}
Bloque 4
−63x+24x=−12−84x+24x=−12−64x=−12−12=−4xx=3\begin{aligned} -\frac{6}{3x} + \frac{2}{4x} &= -\frac{1}{2} \\[10pt] \frac{-8}{4x} + \frac{2}{4x} &= -\frac{1}{2} \\[10pt] \frac{-6}{4x} &= -\frac{1}{2} \\[10pt] -12 &= -4x \\[10pt] \mathbf{x = 3} \end{aligned}
26−12x=53−34x12(13−12x)=12(53−34x)4−6x=20−9x3x=16x=163≈5.33\begin{aligned} \frac{2}{6} - \frac{1}{2}x &= \frac{5}{3} - \frac{3}{4}x \\[10pt] 12 \left(\frac{1}{3} - \frac{1}{2}x\right) &= 12 \left(\frac{5}{3} - \frac{3}{4}x\right) \\[10pt] 4 - 6x &= 20 - 9x \\[10pt] 3x &= 16 \\[10pt] \mathbf{x = \frac{16}{3} \approx 5.33} \end{aligned}
34−52x=35−4(25x−12)34−52x=35−85x+220(34−52x)=20(35−85x+2)15−50x=12−32x+4015−50x=52−32x−37=18xx=−3718≈−2.06\begin{aligned} \frac{3}{4} - \frac{5}{2}x &= \frac{3}{5} - 4\left(\frac{2}{5}x - \frac{1}{2}\right) \\[10pt] \frac{3}{4} - \frac{5}{2}x &= \frac{3}{5} - \frac{8}{5}x + 2 \\[10pt] 20\left(\frac{3}{4} - \frac{5}{2}x\right) &= 20\left(\frac{3}{5} - \frac{8}{5}x + 2\right) \\[10pt] 15 - 50x &= 12 - 32x + 40 \\[10pt] 15 - 50x &= 52 - 32x \\[10pt] -37 &= 18x \\[10pt] \mathbf{x = -\frac{37}{18} \approx -2.06} \end{aligned}

Ejercicios extra

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Bloque 1
−43x−56x=34-\dfrac{4}{3x} - \dfrac{5}{6x} = \dfrac{3}{4}
53x−27=−35x−46\dfrac{5}{3}x - \dfrac{2}{7} = -\dfrac{3}{5}x - \dfrac{4}{6}
2(45+27x)−35x=5x−12\left(\dfrac{4}{5} + \dfrac{2}{7}x\right) - \dfrac{3}{5}x = 5x - 1
Bloque 2
−56x+3x=−25-\dfrac{5}{6x} + \dfrac{3}{x} = -\dfrac{2}{5}
29x−35=3x4−25\dfrac{2}{9}x - \dfrac{3}{5} = \dfrac{3x}{4} - \dfrac{2}{5}
34x−5=65−2(23−45x)\dfrac{3}{4}x - 5 = \dfrac{6}{5} - 2\left(\dfrac{2}{3} - \dfrac{4}{5}x\right)
Bloque 3
−47x+35x=−29-\dfrac{4}{7x} + \dfrac{3}{5x} = -\dfrac{2}{9}
−35+46x=−35x+23-\dfrac{3}{5} + \dfrac{4}{6}x = -\dfrac{3}{5}x + \dfrac{2}{3}
−53x−3(43−2x)=−24x−56-\dfrac{5}{3}x - 3\left(\dfrac{4}{3} - 2x\right) = -\dfrac{2}{4}x - \dfrac{5}{6}
Bloque 4
−25x+12x=−310-\dfrac{2}{5x} + \dfrac{1}{2x} = -\dfrac{3}{10}
43x−12=25x+310\dfrac{4}{3}x - \dfrac{1}{2} = \dfrac{2}{5}x + \dfrac{3}{10}
23−45x=36−4(37x−25)\dfrac{2}{3} - \dfrac{4}{5}x = \dfrac{3}{6} - 4\left(\dfrac{3}{7}x - \dfrac{2}{5}\right)
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