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Math  /  Algebra

QuestionRational Equation with checking  1. 4x−3x6=3x−44\text { 1. } \frac{4 x-3 x}{6}=\frac{3 x-4}{4}

Studdy Solution

STEP 1

1. The given equation is a rational equation involving fractions.
2. To solve the equation, we need to find the value of xx that satisfies the equation.
3. The solution may involve clearing the fractions by finding a common denominator.
4. We need to check the final solution to ensure it doesn't result in any undefined expressions (e.g., division by zero).

STEP 2

1. Simplify the left-hand side of the equation.
2. Clear the fractions by finding a common denominator.
3. Solve the resulting linear equation for xx.
4. Check the solution by substituting it back into the original equation to verify.

STEP 3

Simplify the left-hand side of the equation.
4x−3x6=x6 \frac{4x - 3x}{6} = \frac{x}{6}

STEP 4

Rewrite the equation with the simplified left-hand side.
x6=3x−44 \frac{x}{6} = \frac{3x - 4}{4}

STEP 5

Clear the fractions by finding a common denominator. The least common multiple of 6 and 4 is 12. Multiply both sides by 12.
12⋅x6=12⋅3x−44 12 \cdot \frac{x}{6} = 12 \cdot \frac{3x - 4}{4}

STEP 6

Simplify the equation after multiplying both sides by 12.
2x=3(3x−4) 2x = 3(3x - 4)

STEP 7

Distribute the 3 on the right-hand side.
2x=9x−12 2x = 9x - 12

STEP 8

Solve the linear equation for xx. Start by moving the 9x9x term to the left-hand side.
2x−9x=−12 2x - 9x = -12

STEP 9

Combine like terms.
−7x=−12 -7x = -12

STEP 10

Divide both sides by −7-7 to isolate xx.
x=−12−7 x = \frac{-12}{-7}

STEP 11

Simplify the division.
x=127 x = \frac{12}{7}

STEP 12

Check the solution by substituting x=127x = \frac{12}{7} back into the original equation.
4(127)−3(127)6=3(127)−44 \frac{4 \left( \frac{12}{7} \right) - 3 \left( \frac{12}{7} \right)}{6} = \frac{3 \left( \frac{12}{7} \right) - 4}{4}

STEP 13

Simplify the left-hand side with x=127x = \frac{12}{7}.
487−3676=1276=127⋅16=1242=27 \frac{\frac{48}{7} - \frac{36}{7}}{6} = \frac{\frac{12}{7}}{6} = \frac{12}{7} \cdot \frac{1}{6} = \frac{12}{42} = \frac{2}{7}

STEP 14

Simplify the right-hand side with x=127x = \frac{12}{7}.
3(127)−44=367−44=367−2874=874=87⋅14=828=27 \frac{3 \left( \frac{12}{7} \right) - 4}{4} = \frac{\frac{36}{7} - 4}{4} = \frac{\frac{36}{7} - \frac{28}{7}}{4} = \frac{\frac{8}{7}}{4} = \frac{8}{7} \cdot \frac{1}{4} = \frac{8}{28} = \frac{2}{7}

STEP 15

Since both sides are equal when x=127x = \frac{12}{7}, the solution satisfies the original equation.
Therefore, the solution is:
x=127 x = \frac{12}{7}

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