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How to Solve the Linear Equation 4(3x + 2) - x = 7x + 2(2x + 4)
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the linear equation: 4(3x + 2) - x = 7x + 2(2x + 4)
Correct Answer
All real numbers (infinitely many solutions)
Detailed Solution Steps
1
Step 1: Expand both sides of the equation using the distributive property. On the left side: 4(3x+2)-x = 12x + 8 - x; On the right side: 7x + 2(2x+4) = 7x + 4x + 8
2
Step 2: Combine like terms on each side. Left side simplifies to: 12x - x + 8 = 11x + 8; Right side simplifies to: 7x + 4x + 8 = 11x + 8
3
Step 3: Subtract 11x + 8 from both sides to isolate the variable. This results in 0 = 0, which is a true statement for all values of x.
Knowledge Points Involved
1
Distributive Property of Multiplication over Addition
Also called the distributive law, it states that a(b + c) = ab + ac. This property is used to expand expressions with parentheses, allowing us to eliminate grouping symbols in linear equations.
2
Combining Like Terms
Like terms are terms with the same variable raised to the same power. We combine them by adding or subtracting their coefficients, which simplifies algebraic expressions and linear equations.
3
Types of Solutions for Linear Equations
A linear equation can have one unique solution, no solution, or infinitely many solutions. If simplifying leads to a true statement (e.g., 0=0), the equation has infinitely many solutions (all real numbers are valid).
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