Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar. What value of $p$ makes the equation true? [tex]$-3 p+\frac{1}{8}=-\frac{1}{4}$[/tex] $p=$
Answer (2)
Isolate the term with p by subtracting 8 1 from both sides: − 3 p = − 4 1 − 8 1 .
Simplify the right side by finding a common denominator: − 3 p = − 8 2 − 8 1 = − 8 3 .
Solve for p by dividing both sides by − 3 : p = − 3 − 8 3 .
Simplify to find the value of p : p = 8 1 .
8 1
Explanation
Problem Analysis We are given the equation − 3 p + 8 1 = − f r a c 1 4 and we need to find the value of p that makes this equation true.
Isolating the p term First, we want to isolate the term with p . To do this, we subtract f r a c 1 8 from both sides of the equation: − 3 p + f r a c 1 8 − f r a c 1 8 = − f r a c 1 4 − f r a c 1 8 − 3 p = − f r a c 1 4 − f r a c 1 8
Finding a common denominator Next, we simplify the right side of the equation by finding a common denominator for the fractions. The common denominator for 4 and 8 is 8. So we rewrite − f r a c 1 4 as − f r a c 2 8 : − 3 p = − f r a c 2 8 − f r a c 1 8
Combining fractions Now, we combine the fractions on the right side: − 3 p = − f r a c 2 8 − f r a c 1 8 = − f r a c 3 8
Solving for p To solve for p , we divide both sides of the equation by − 3 :
p = f r a c − f r a c 3 8 − 3
Simplifying the expression Now we simplify the expression: p = f r a c − f r a c 3 8 − 3 = f r a c 3 8 c d o t f r a c 1 3 = f r a c 3 24
Final simplification Finally, we simplify the fraction: p = f r a c 3 24 = f r a c 1 8
Final Answer Therefore, the value of p that makes the equation true is f r a c 1 8 .
Examples
Understanding how to solve linear equations is crucial in many real-world scenarios. For instance, imagine you're managing a budget for a project. You know that you have a certain amount of money to spend, and you need to allocate it to different areas. If you know the cost of some items and the total budget, you can set up a linear equation to determine how much you can spend on other items. For example, if your total budget is $1000, and you've already spent $200 on materials and $300 on labor, you can use the equation 200 + 300 + x = 1000 to find out how much money ( x ) you have left for marketing. Solving for x gives you $x = 1000 - 200 - 300 = $500. This shows how solving linear equations helps in managing budgets and making financial decisions.
To solve the equation − 3 p + 8 1 = − 4 1 , we isolate p by finding a common denominator, combining fractions, and then solving for p . The final answer is p = 8 1 .
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