The equation $8(3 x-2)-4=8 x+4(4 x-3)$ has what type of solution set?
A. Two-solution
B. No solution
C. One-solution
D. Infinite
Answer (2)
Expand both sides of the equation: 8 ( 3 x − 2 ) − 4 = 8 x + 4 ( 4 x − 3 ) becomes 24 x − 20 = 24 x − 12 .
Simplify the equation by subtracting 24 x from both sides: − 20 = − 12 .
Since − 20 = − 12 is a false statement, there is no value of x that satisfies the equation.
Therefore, the equation has no solution: No solution .
Explanation
Analyze the problem We are given the equation 8 ( 3 x − 2 ) − 4 = 8 x + 4 ( 4 x − 3 ) and asked to determine the type of solution set it has. The possible solution sets are: two-solution, no solution, one-solution, or infinite solutions. To determine the type of solution set, we need to simplify the equation and solve for x .
Expand both sides First, expand both sides of the equation: 8 ( 3 x − 2 ) − 4 = 8 x + 4 ( 4 x − 3 ) 24 x − 16 − 4 = 8 x + 16 x − 12 24 x − 20 = 24 x − 12
Simplify both sides Next, simplify both sides of the equation by combining like terms. In this case, the x terms are already combined on both sides, and the constant terms are also combined.
Isolate the variable Now, subtract 24 x from both sides of the equation: 24 x − 20 − 24 x = 24 x − 12 − 24 x − 20 = − 12
Analyze the result The equation simplifies to − 20 = − 12 , which is a false statement. This means that there is no value of x that will satisfy the original equation.
Conclusion Therefore, the equation has no solution.
Examples
Understanding the types of solutions an equation can have is crucial in many real-world applications. For instance, when designing a bridge, engineers use equations to model the forces and stresses acting on the structure. If an equation representing a critical aspect of the design has no solution, it indicates a fundamental flaw that must be addressed to ensure the bridge's stability. Similarly, in economics, if a model predicting market equilibrium yields no solution, it suggests that the assumptions underlying the model are inconsistent with the observed data. Recognizing the absence of a solution is therefore a vital step in problem-solving across various disciplines.
The equation 8 ( 3 x − 2 ) − 4 = 8 x + 4 ( 4 x − 3 ) simplifies to a false statement − 20 = − 12 , indicating that there are no solutions. Therefore, the correct answer is B. No solution.
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