Identify the equation as a conditional equation, a contradiction, or an identity.
[tex]5(t+7)-6=59[/tex]
Part 1 of 2
The equation is a conditional equation.
The equation is a contradiction.
The equation is an identity.
Answer (2)
The equation 5 ( t + 7 ) − 6 = 59 is a conditional equation, as it is true only for the specific value t = 6 .
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Simplify the equation: 5 ( t + 7 ) − 6 = 5 t + 29 .
Set up the simplified equation: 5 t + 29 = 59 .
Solve for t : 5 t = 30 , so t = 6 .
Since the equation is true only for t = 6 , it is a conditional equation.
The equation is a conditional equation.
Explanation
Understanding the Problem We are given the equation 5 ( t + 7 ) − 6 = 59 and asked to identify whether it is 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 left-hand side of the equation by distributing the 5: 5 ( t + 7 ) − 6 = 5 t + 35 − 6 = 5 t + 29
Isolating the Variable Now we have the equation 5 t + 29 = 59 . We want to isolate the variable t . Subtract 29 from both sides of the equation: 5 t + 29 − 29 = 59 − 29
5 t = 30
Solving for t Divide both sides by 5: 5 5 t = 5 30
t = 6
Conclusion Since we found a specific value for t that makes the equation true ( t = 6 ), the equation is a conditional equation.
Examples
Conditional equations are useful in many real-world scenarios. For example, suppose you want to save up for a new bicycle that costs $300. You already have $50 saved, and you plan to save $25 each week. The equation 50 + 25 w = 300 is a conditional equation that represents this situation, where w is the number of weeks. Solving for w will tell you how many weeks it will take to save enough money to buy the bicycle. In this case, w = 10 , so it will take 10 weeks to save enough money.
