Select the correct answer.
Which equation has no solution?
A. $-4(y+7)=2(-2 y-9)-10$
B. $-2(2 y+8)=4 y+5+y$
C. $4(y+9)=-4(y-9)$
D. $3 y+5-7 y=4(-y+1)+5$
Answer (2)
Simplify each equation.
Equation A simplifies to an identity, meaning it has infinite solutions.
Equations B and C simplify to a unique solution for y .
Equation D simplifies to a contradiction ( 5 = 9 ), indicating no solution. Therefore, the answer is D .
Explanation
Understanding the Problem We are given four linear equations and asked to identify the one with no solution. An equation has no solution if, after simplification, we arrive at a contradiction (e.g., 0 = 1 ).
Simplifying and Analyzing Each Equation Let's simplify each equation to determine if it has a solution or leads to a contradiction.
Equation A: − 4 ( y + 7 ) = 2 ( − 2 y − 9 ) − 10 Expanding both sides, we get: − 4 y − 28 = − 4 y − 18 − 10 − 4 y − 28 = − 4 y − 28 Adding 4 y to both sides, we get: − 28 = − 28 This equation is always true, so it has infinitely many solutions.
Equation B: − 2 ( 2 y + 8 ) = 4 y + 5 + y Expanding the left side, we get: − 4 y − 16 = 5 y + 5 Adding 4 y to both sides, we get: − 16 = 9 y + 5 Subtracting 5 from both sides, we get: − 21 = 9 y Dividing by 9, we get: y = − 9 21 = − 3 7 This equation has one solution.
Equation C: 4 ( y + 9 ) = − 4 ( y − 9 ) Expanding both sides, we get: 4 y + 36 = − 4 y + 36 Adding 4 y to both sides, we get: 8 y + 36 = 36 Subtracting 36 from both sides, we get: 8 y = 0 Dividing by 8, we get: y = 0 This equation has one solution.
Equation D: 3 y + 5 − 7 y = 4 ( − y + 1 ) + 5 Simplifying the left side, we get: − 4 y + 5 = − 4 y + 4 + 5 − 4 y + 5 = − 4 y + 9 Adding 4 y to both sides, we get: 5 = 9 This is a contradiction, so the equation has no solution.
Identifying the Equation with No Solution Therefore, equation D has no solution because it simplifies to a contradiction ( 5 = 9 ).
Examples
Understanding equations with no solution is crucial in various fields. For instance, in engineering, if a system of equations describing a circuit has no solution, it indicates a design flaw or an impossible configuration. Similarly, in economics, if a model predicts a scenario where supply and demand equations have no solution, it suggests market instability or an unrealistic model. Recognizing such situations allows for adjustments and corrections to ensure practical and meaningful outcomes.
Upon evaluating all four equations, Equation D simplifies to a contradiction ( 5 = 9 ), indicating that it has no solution. In contrast, Equations A, B, and C either yield infinitely many solutions or a unique solution, respectively. Consequently, the answer is D.
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