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How to Solve the Rational Equation $4 = \\frac{3x - 7}{3x}$ for x
Mathematics
Grade 9 (Junior High School)
Question Content
Solve for x: $4 = \\frac{3x - 7}{3x}$
Correct Answer
$x = -\\frac{7}{9}$
Detailed Solution Steps
1
Step 1: Eliminate the denominator by multiplying both sides of the equation by $3x$: $4 \\times 3x = 3x - 7$
2
Step 2: Simplify the left side of the equation: $12x = 3x - 7$
3
Step 3: Isolate the terms with $x$ by subtracting $3x$ from both sides: $12x - 3x = -7$
4
Step 4: Combine like terms: $9x = -7$
5
Step 5: Solve for $x$ by dividing both sides by 9: $x = -\\frac{7}{9}$
6
Step 6: Verify the solution: Substitute $x=-\\frac{7}{9}$ back into the original equation, the denominator $3x = 3\\times(-\\frac{7}{9}) = -\\frac{7}{3} \\neq 0$, and the right-hand side $\\frac{3\\times(-\\frac{7}{9}) -7}{3\\times(-\\frac{7}{9})} = \\frac{-\\frac{7}{3}-7}{-\\frac{7}{3}} = \\frac{-\\frac{28}{3}}{-\\frac{7}{3}} = 4$, which matches the left-hand side, so the solution is valid.
Knowledge Points Involved
1
Rational Equation Solving
A rational equation is an equation containing rational expressions (fractions with polynomials in numerator/denominator). The core method is to eliminate denominators by multiplying both sides by the least common denominator, then solve the resulting linear/quadratic equation, and always check for extraneous solutions that make any original denominator zero.
2
Isolating Variables
This is a foundational algebra skill where you use inverse operations (addition/subtraction, multiplication/division) to rearrange an equation so that the variable you are solving for is alone on one side of the equals sign. It applies to linear, rational, and other types of equations.
3
Combining Like Terms
Like terms are terms with the same variable raised to the same power. Combining them involves adding or subtracting their coefficients to simplify algebraic expressions or equations, which is a key step in solving most algebraic equations.
4
Solution Verification
After solving an equation, especially rational equations, you must substitute the solution back into the original equation to ensure it makes the equation true and does not result in a division by zero. This step catches extraneous solutions introduced when eliminating denominators.
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