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Solve Rational Equation 7/(x - 4) + 3/4 = 4
Mathematics
Junior High School (Grade 7-9)
Question Content
Solve the equation \\( \\dfrac{7}{x - 4} + \\dfrac{3}{4} = 4 \\)
Correct Answer
x = \\( \\dfrac{80}{13} \\)
Detailed Solution Steps
1
Step 1: Eliminate denominators by multiplying both sides by \( 4(x - 4) \) (the least common multiple of \( x - 4 \) and \( 4 \)) to avoid fractions:
2
\( 4(x - 4) \\cdot \\left( \\dfrac{7}{x - 4} + \\dfrac{3}{4} \\right) = 4 \\cdot 4(x - 4) \)
3
Step 2: Distribute and simplify the left side. The \( (x - 4) \) cancels in the first term, and the \( 4 \) cancels in the second term:
4
\( 4(x - 4) \\cdot \\dfrac{7}{x - 4} + 4(x - 4) \\cdot \\dfrac{3}{4} = 16(x - 4) \)
5
\( 28 + 3(x - 4) = 16(x - 4) \)
6
Step 3: Expand both sides:
7
\( 28 + 3x - 12 = 16x - 64 \)
8
Step 4: Simplify the left side (combine like terms) and the right side:
9
\( 16 + 3x = 16x - 64 \)
10
Step 5: Subtract \( 3x \) from both sides to isolate \( x \) terms on the right:
11
\( 16 = 13x - 64 \)
12
Step 6: Add \( 64 \) to both sides to isolate the \( x \) term:
13
\( 80 = 13x \)
14
Step 7: Divide both sides by \( 13 \) to solve for \( x \):
15
\( x = \\dfrac{80}{13} \)
Knowledge Points Involved
1
Solving Rational Equations
Rational equations contain fractions with variables in the denominator. To solve them, eliminate denominators by multiplying both sides by the least common denominator (LCD) of all fractions. This avoids division by zero (so check solutions for validity).
2
Least Common Denominator (LCD)
The LCD of two or more denominators is the smallest expression divisible by each denominator. For \( x - 4 \) and \( 4 \), the LCD is \( 4(x - 4) \), used here to eliminate fractions.
3
Algebraic Manipulation (Expanding, Combining Like Terms)
Expanding involves distributing a factor across a sum (e.g., \( a(b + c) = ab + ac \)). Combining like terms simplifies expressions by adding/subtracting coefficients of the same variable or constant terms.
4
Checking Solutions
After solving a rational equation, substitute the solution back into the original equation to ensure it does not make any denominator zero (validity) and satisfies the equation (accuracy).
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