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Step-by-Step Solution for the Fractional One-Variable Linear Equation $\\frac{x}{7}-\\frac{5}{21}=\\frac{x}{3}+\\frac{2}{7}$
Mathematics
Grade 7 of Junior High School
Question Content
Solve the equation: $\\frac{x}{7}-\\frac{5}{21}=\\frac{x}{3}+\\frac{2}{7}$
Correct Answer
$x=-11/4$
Detailed Solution Steps
1
Step 1: Find the LCM of 7, 21 and 3, which is 21. Multiply every term in the equation by 21: $21\\times\\frac{x}{7} - 21\\times\\frac{5}{21} = 21\\times\\frac{x}{3} + 21\\times\\frac{2}{7}$
2
Step 2: Simplify each term: $3x - 5 = 7x + 6$
3
Step 3: Transpose terms to group $x$ terms on one side and constants on the other: $3x - 7x = 6 + 5$
4
Step 4: Combine like terms: $-4x = 11$
5
Step 5: Solve for $x$ by dividing both sides by -4: $x=-11/4$
Knowledge Points Involved
1
Linear Equations with One Variable (Fractional Form)
A one-variable linear equation that contains fractional coefficients or constant terms. The core solving method is to convert it into an integer coefficient equation first, then solve it using basic equality properties.
2
Least Common Multiple (LCM) of Multiple Numbers
For multiple positive integers, the LCM is the smallest positive integer that is divisible by all of them. When dealing with denominators like 7, 3 and 21, we can identify the LCM by observing that 21 is a multiple of both 7 and 3, so 21 is the LCM here.
3
Property of Equality of Equations
If $a=b$, then $a\\times c = b\\times c$ and $a\\div c = b\\div c$ (where $c\\neq0$). This property allows us to multiply each term of the equation by the same non-zero number to eliminate denominators without changing the solution of the equation.
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