What is the value of $p$ in the linear equation $24 p+12-18 p=10+2 p-6?
Answer (1)
Simplify the equation by combining like terms: 6 p + 12 = 2 p + 4 .
Isolate the variable p by subtracting 2 p and 12 from both sides: 4 p = − 8 .
Solve for p by dividing both sides by 4 : p = − 2 .
Verify the solution by substituting p = − 2 back into the original equation. The final answer is − 2 .
Explanation
Simplifying the equation First, we need to simplify the given linear equation: 24 p + 12 − 18 p = 10 + 2 p − 6 Combine the like terms on both sides of the equation. On the left side, we combine the terms with p : 24 p − 18 p = 6 p . So the left side becomes 6 p + 12 . On the right side, we combine the constant terms: 10 − 6 = 4 . So the right side becomes 2 p + 4 . Now the equation is: 6 p + 12 = 2 p + 4
Isolating the variable Next, we want to isolate the terms with p on one side of the equation and the constant terms on the other side. Subtract 2 p from both sides: 6 p − 2 p + 12 = 2 p − 2 p + 4 4 p + 12 = 4 Now, subtract 12 from both sides: 4 p + 12 − 12 = 4 − 12 4 p = − 8
Solving for p Now, we solve for p by dividing both sides by 4: 4 4 p = 4 − 8 p = − 2
Verifying the solution Finally, we verify the solution by substituting p = − 2 back into the original equation: 24 ( − 2 ) + 12 − 18 ( − 2 ) = 10 + 2 ( − 2 ) − 6 − 48 + 12 + 36 = 10 − 4 − 6 − 48 + 48 = 10 − 10 0 = 0 The solution is correct.
Final Answer Therefore, the value of p in the linear equation is − 2 .
Examples
Linear equations are used in various real-life scenarios, such as calculating the cost of items with discounts or taxes. For example, suppose a store offers a 20% discount on an item, and there's a sales tax of 8%. If the final price you pay is $32.40, you can set up a linear equation to find the original price of the item before the discount and tax. Understanding how to solve linear equations helps in managing personal finances, making informed purchasing decisions, and understanding basic economic principles.
