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How to Solve the One-Variable Linear Equation $\\frac{x}{2}+4=\\frac{x}{5}-1$
Mathematics
Grade 7 of Junior High School
Question Content
Solve the equation: $\\frac{x}{2}+4=\\frac{x}{5}-1$
Correct Answer
$x=-50/3$
Detailed Solution Steps
1
Step 1: Eliminate the denominators by finding the least common multiple (LCM) of 2 and 5, which is 10. Multiply every term in the equation by 10: $10\\times\\frac{x}{2} + 10\\times4 = 10\\times\\frac{x}{5} - 10\\times1$
2
Step 2: Simplify each term after multiplication: $5x + 40 = 2x - 10$
3
Step 3: Move all terms with $x$ to the left side and constant terms to the right side: $5x - 2x = -10 - 40$
4
Step 4: Combine like terms: $3x = -50$
5
Step 5: Solve for $x$ by dividing both sides by 3: $x=-50/3$
Knowledge Points Involved
1
Linear Equations with One Variable
An equation that contains only one unknown variable, and the highest power of the variable is 1. Its standard form is $ax+b=0$ ($a\\neq0$), and it has exactly one solution, no solution, or infinitely many solutions depending on the coefficients.
2
Least Common Multiple (LCM) for Eliminating Denominators
When solving fractional linear equations, we multiply each term by the LCM of all denominators to convert the equation into an integer coefficient equation, which simplifies calculation. For two integers, the LCM can be found by factoring or using the formula $LCM(a,b)=\\frac{|a\\times b|}{GCD(a,b)}$ where GCD is the greatest common divisor.
3
Transposition of Terms in Equations
When moving a term from one side of an equation to the other, its sign must be changed. This is based on the property of equality: adding or subtracting the same number or term on both sides of an equation does not change the equality.
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