Solve for $k$. $8 k+2 m=3 m+k$
Answer (2)
Simplify equation (1) to express k in terms of m : k = 7 m .
Compare this result with equations (2), (3), (4), and (5) to find possible values for m and k .
Analyze the cases where m = 0 , m = 7 , and m = − 7 , checking if they satisfy all equations.
Conclude that there is no solution for k that satisfies all equations simultaneously, as equations (2) and (3) are not consistently satisfied. Therefore, there is no solution. no solution
Explanation
Understanding the Problem We are given the following equations:
8 k + 2 m = 3 m + k
k = 70 m
k = 7 m
k = m 7
k = 7 m
Our goal is to solve for k .
Simplifying Equation (1) First, let's simplify equation (1):
8 k + 2 m = 3 m + k 7 k = m k = 7 m
Comparing with Other Equations Now we have k = 7 m . Let's compare this with the other given equations.
Comparing with equation (2): k = 70 m . If k = 7 m , then 7 m = 70 m . This implies m = 490 m , which means 489 m = 0 , so m = 0 . If m = 0 , then k = 0 .
Comparing with equation (3): k = 7 m . If k = 7 m , then 7 m = 7 m . This implies m = 49 m , which means 48 m = 0 , so m = 0 . If m = 0 , then k = 0 .
Comparing with equation (4): k = m 7 . If k = 7 m , then 7 m = m 7 . This implies m 2 = 49 , so m = ± 7 . If m = 7 , then k = 1 . If m = − 7 , then k = − 1 .
Comparing with equation (5): k = 7 m . This is consistent with the simplified equation from (1).
Analyzing Possible Solutions Let's analyze the possible solutions:
Case 1: m = 0 and k = 0 .
From equation (4), k = m 7 is undefined, so m cannot be 0.
Case 2: m = 7 and k = 1 .
Equation (1): 8 ( 1 ) + 2 ( 7 ) = 3 ( 7 ) + 1 , which simplifies to 8 + 14 = 21 + 1 , or 22 = 22 . This is true. Equation (2): 1 = 70 ( 7 ) , which is false. Equation (3): 1 = 7 ( 7 ) , which is false. Equation (4): 1 = 7 7 , which is true. Equation (5): 1 = 7 7 , which is true.
Case 3: m = − 7 and k = − 1 .
Equation (1): 8 ( − 1 ) + 2 ( − 7 ) = 3 ( − 7 ) + ( − 1 ) , which simplifies to − 8 − 14 = − 21 − 1 , or − 22 = − 22 . This is true. Equation (2): − 1 = 70 ( − 7 ) , which is false. Equation (3): − 1 = 7 ( − 7 ) , which is false. Equation (4): − 1 = − 7 7 , which is true. Equation (5): − 1 = 7 − 7 , which is true.
Conclusion Since equations (2) and (3) are not satisfied by either solution, there is no solution for k that satisfies all equations simultaneously.
Examples
In electrical engineering, you might have a circuit where the voltage (k) and current (m) are related by several equations due to different components and constraints. Solving this system of equations ensures that the circuit operates as intended, with the correct voltage and current levels at various points. If the equations are inconsistent, it indicates a problem in the circuit design or component selection, which needs to be addressed to ensure proper functionality.
To solve for k in the equation 8 k + 2 m = 3 m + k , we simplify it to find that k = 7 m . This means that for any given value of m , we can calculate k directly. For example, if m = 7 , then k = 1 and if m = − 7 , then k = − 1 .
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