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Solve the Multi-Step Linear Equation 4(3x + 5) - 2x = -2 + 2(5x + 11)
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the multi-step equation: 4(3x + 5) - 2x = -2 + 2(5x + 11)
Correct Answer
Infinitely many solutions (all real numbers are solutions)
Detailed Solution Steps
1
Step 1: Expand both sides of the equation using the distributive property. On the left side: 4(3x + 5) - 2x = 12x + 20 - 2x. On the right side: -2 + 2(5x + 11) = -2 + 10x + 22
2
Step 2: Combine like terms on each side. Left side: 12x - 2x + 20 = 10x + 20. Right side: 10x + (-2 + 22) = 10x + 20
3
Step 3: Subtract 10x + 20 from both sides of the equation. This results in 0 = 0, which is a true statement for all real values of x.
Knowledge Points Involved
1
Distributive Property of Multiplication
The property states that a(b + c) = ab + ac. It is used to expand expressions with parentheses, allowing us to eliminate grouping symbols and simplify algebraic equations. It applies to all real numbers a, b, and c.
2
Combining Like Terms
Like terms are terms with the same variable raised to the same power. To combine them, add or subtract their coefficients while keeping the variable part unchanged. This simplifies algebraic expressions and equations to make them easier to solve.
3
Classifying Solutions to Linear Equations
When solving linear equations: if the result is a true statement (e.g., 0=0), the equation has infinitely many solutions; if the result is a false statement (e.g., 0=5), the equation has no solution; if the result isolates the variable (e.g., x=3), the equation has exactly one solution.
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