Solve [tex]\frac{3-5 m}{-6+10 m}=\frac{7+19 k}{4 k}[/tex]
Answer (1)
k = − 3 1 and m = 5 3 .
k = − 3 1 and m = 5 3
Explanation
Analyzing the Problem We are given the equation − 6 + 10 m 3 − 5 m = 4 k 7 + 19 k and we want to find the relationship between m and k .
Simplifying the Left-Hand Side First, notice that the denominator on the left-hand side can be written as − 6 + 10 m = − 2 ( 3 − 5 m ) So the left-hand side simplifies to − 2 ( 3 − 5 m ) 3 − 5 m
Further Simplification We can simplify the fraction on the left-hand side, assuming 3 − 5 m = 0 , which means m = 5 3 . Then we have − 2 ( 3 − 5 m ) 3 − 5 m = − 2 1 So the equation becomes − 2 1 = 4 k 7 + 19 k .
Solving for k Now, we solve for k . Multiply both sides of the equation by 4 k :
− 2 1 ( 4 k ) = 7 + 19 k − 2 k = 7 + 19 k Subtract 19 k from both sides: − 21 k = 7 Divide by − 21 :
k = − 21 7 = − 3 1 .
Checking Consistency Now we need to check if this value of k is consistent with the original equation. Substitute k = − 3 1 into the original equation: − 6 + 10 m 3 − 5 m = 4 ( − 3 1 ) 7 + 19 ( − 3 1 ) − 6 + 10 m 3 − 5 m = − 3 4 7 − 3 19 = − 3 4 3 21 − 19 = − 3 4 3 2 = − 4 2 = − 2 1 So we have − 6 + 10 m 3 − 5 m = − 2 1 − 2 ( 3 − 5 m ) 3 − 5 m = − 2 1 This holds for all m such that 3 − 5 m = 0 , which means m = 5 3 .
Final Answer Therefore, the solution is k = − 3 1 and m = 5 3 .
Examples
This problem demonstrates how to solve equations involving fractions and multiple variables. In real life, such equations can model relationships between different quantities, such as the relationship between production costs and sales revenue, or the relationship between the flow rates of different pipes in a network. By solving these equations, we can determine the values of the variables that satisfy the given conditions, which can help us make informed decisions in various fields, such as business, engineering, and science. For example, if k represents the interest rate and m represents the inflation rate, solving the equation can help us understand how these two factors affect investment returns.
