Identify the equation as a conditional equation, contradiction, or an identity. Then give the solution.
[tex]3 m+9-12 m=-4 m+6-9 m[/tex]
Answer (2)
The equation 3 m + 9 − 12 m = − 4 m + 6 − 9 m simplifies to m = − 4 3 , indicating that it is a conditional equation. This type of equation has one specific solution. Thus, m = − 4 3 is valid for this equation.
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Simplify both sides of the equation by combining like terms: − 9 m + 9 = − 13 m + 6 .
Add 13 m to both sides: 4 m + 9 = 6 .
Subtract 9 from both sides: 4 m = − 3 .
Divide by 4 to find the value of m : m = − 4 3 .
Explanation
Understanding the Problem We are given the equation 3 m + 9 − 12 m = − 4 m + 6 − 9 m . Our goal is to determine whether this equation is a conditional equation, a contradiction, or an identity.
Simplifying Both Sides First, we simplify both sides of the equation by combining like terms. On the left side, we have 3 m − 12 m + 9 , which simplifies to − 9 m + 9 . On the right side, we have − 4 m − 9 m + 6 , which simplifies to − 13 m + 6 .
Isolating the Variable Now our equation is − 9 m + 9 = − 13 m + 6 . To solve for m , we want to isolate m on one side of the equation. We can add 13 m to both sides of the equation: − 9 m + 13 m + 9 = − 13 m + 13 m + 6 This simplifies to 4 m + 9 = 6 .
Further Isolation Next, we subtract 9 from both sides of the equation: 4 m + 9 − 9 = 6 − 9 This simplifies to 4 m = − 3 .
Solving for m Finally, we divide both sides by 4 to solve for m :
m = 4 − 3 So, m = − 4 3 .
Conclusion Since the equation is true for only one specific value of m , namely m = − 4 3 , the equation is a conditional equation.
Examples
Conditional equations are useful in many real-world scenarios. For example, consider a situation where you want to determine the number of items you need to sell to break even. If your cost function is C ( x ) = 5 x + 100 and your revenue function is R ( x ) = 10 x , you can set C ( x ) = R ( x ) to find the break-even point. This gives you the conditional equation 5 x + 100 = 10 x . Solving for x gives x = 20 , meaning you need to sell 20 items to break even. This is a conditional equation because it is only true for x = 20 .
