Which expression is missing from step 7?
8. [tex]2+ d ^2+ e ^2= d ^2-2 d e + e ^2[/tex]
[tex]2=-2 d e[/tex]
[tex]-1=d e[/tex]
[tex](1+d^2)+(e^2+1)=d^2-2 d e+e^2[/tex]
substitution property of equaling
Answer (1)
The equation in step 8 is 2 + d 2 + e 2 = d 2 − 2 d e + e 2 .
The equation given is ( 1 + d 2 ) + ( e 2 + 1 ) = d 2 − 2 d e + e 2 .
The missing expression involves splitting the constant 2 into 1+1.
The missing expression is 1 + d 2 + e 2 + 1 = d 2 − 2 d e + e 2 .
Explanation
Understanding the Problem We are given step 8: 2 + d 2 + e 2 = d 2 − 2 d e + e 2 which simplifies to 2 = − 2 d e and − 1 = d e . We are also given ( 1 + d 2 ) + ( e 2 + 1 ) = d 2 − 2 d e + e 2 which is obtained from the previous step using the substitution property of equality. We need to find the expression that is missing from step 7.
Analyzing the Equations The equation in step 8, 2 + d 2 + e 2 = d 2 − 2 d e + e 2 , can be rewritten as 2 + d 2 + e 2 = ( d − e ) 2 . The equation given, ( 1 + d 2 ) + ( e 2 + 1 ) = d 2 − 2 d e + e 2 , can be rewritten as 1 + d 2 + e 2 + 1 = ( d − e ) 2 . This suggests that the missing expression involves splitting the constant 2 into 1+1.
Deducing the Missing Expression Since the equation in step 8 is 2 + d 2 + e 2 = d 2 − 2 d e + e 2 and the equation in the problem is ( 1 + d 2 ) + ( e 2 + 1 ) = d 2 − 2 d e + e 2 , we can see that the left side of the equation in step 8 has 2 + d 2 + e 2 and the equation given has 1 + d 2 + 1 + e 2 . This suggests that the missing expression involves splitting the constant 2 into 1+1. Therefore, the missing expression in step 7 is the original equation before the substitution property of equality was applied.
Final Answer The missing expression is 1 + d 2 + e 2 + 1 = d 2 − 2 d e + e 2 .
Examples
Understanding algebraic manipulations is crucial in various fields, such as engineering, physics, and computer science. For instance, when designing a bridge, engineers use algebraic equations to model the forces acting on the structure and ensure its stability. By manipulating these equations, they can optimize the design and prevent potential failures. Similarly, in computer graphics, algebraic equations are used to create realistic images and animations. By understanding how to manipulate these equations, developers can create more immersive and visually appealing experiences.
