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 (1)
Simplify equation A and find it has infinite solutions.
Simplify equation B and find y = 3 − 7 .
Simplify equation C and find y = 0 .
Simplify equation D and find a contradiction, meaning no solution. The answer is D .
Explanation
Problem Analysis We are given four linear equations and asked to identify the one with no solution. To do this, we will simplify each equation and solve for y . An equation has no solution if, after simplification, we arrive at a contradiction (e.g., 0 = 1 ).
Solving Equation A Equation A: − 4 ( y + 7 ) = 2 ( − 2 y − 9 ) − 10 . Let's simplify: − 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, regardless of the value of y . This means that the equation has infinitely many solutions, not no solution.
Solving Equation B Equation B: − 2 ( 2 y + 8 ) = 4 y + 5 + y . Let's simplify: − 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, y = 3 − 7 .
Solving Equation C Equation C: 4 ( y + 9 ) = − 4 ( y − 9 ) . Let's simplify: 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, y = 0 .
Solving Equation D Equation D: 3 y + 5 − 7 y = 4 ( − y + 1 ) + 5 . Let's simplify: − 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, meaning that there is no value of y that satisfies this equation. Therefore, equation D has no solution.
Final Answer Therefore, the equation with no solution is D.
Conclusion The equation that has no solution is D.
Examples
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 in the design that must be corrected to ensure the bridge's stability and safety. Similarly, in economics, if a model predicts supply and demand equilibrium but the related equation has no solution, it suggests an imbalance in the market that requires intervention.
