Identify the equation as a conditional equation, contradiction, or identity.
[tex]5 m+9=-2(-4 m-3)-5[/tex]
Answer (2)
The equation 5 m + 9 = − 2 ( − 4 m − 3 ) − 5 is classified as a conditional equation because it has a unique solution for m = 3 8 . This means the equation is true only for this specific value of m . Therefore, it is not impossible or always true, making it conditional.
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Distribute and simplify the equation: 5 m + 9 = − 2 ( − 4 m − 3 ) − 5 becomes 5 m + 9 = 8 m + 1 .
Isolate the variable m : Subtract 5 m from both sides to get 9 = 3 m + 1 , then subtract 1 from both sides to get 8 = 3 m .
Solve for m : Divide both sides by 3 to find m = 3 8 .
Conclude that the equation is conditional because it is only true for m = 3 8 .
Explanation
Understanding the Problem We are given the equation 5 m + 9 = − 2 ( − 4 m − 3 ) − 5 and asked to classify it as a conditional equation, a contradiction, or an identity. A conditional equation is true for only some values of the variable. A contradiction is never true, no matter the value of the variable. An identity is always true, no matter the value of the variable.
Simplifying the Equation First, we simplify the right side of the equation by distributing the − 2 :
5 m + 9 = 8 m + 6 − 5
Combining Like Terms Next, we combine like terms on the right side:
5 m + 9 = 8 m + 1
Isolating the Variable Now, we want to isolate the variable m . We can subtract 5 m from both sides:
9 = 3 m + 1
Further Isolating the Variable Next, subtract 1 from both sides:
8 = 3 m
Solving for m Finally, divide both sides by 3 to solve for m :
m = 3 8
Conclusion Since we found a unique value for m that satisfies the equation, the equation is a conditional equation.
Examples
Conditional equations are useful in many real-world scenarios. For example, suppose you want to determine how many hours you need to work at a job that pays $15 per hour to earn enough money to buy a new phone that costs $240. The equation would be 15 h = 240 , where h is the number of hours you need to work. Solving for h , you find that h = 16 hours. This is a conditional equation because it is only true when h = 16 .
