Which of the following indicates the subtraction property of equality when solving the equation [tex]$4(x+9)+5=3 x-8$[/tex]?
A. [tex]$4 x+36+5=3 x-8$[/tex]
B. [tex]$x=-49$[/tex]
C. [tex]$4(x+9)-5+8=3 x$[/tex]
D. [tex]$4(x+9)=3 x-8-5$[/tex]
Answer (1)
The subtraction property of equality states that subtracting the same value from both sides of an equation maintains the equality.
Examine each option to see if it results from subtracting a constant from both sides of the original equation.
Option 4 ( x + 9 ) = 3 x − 8 − 5 is obtained by subtracting 5 from both sides of the original equation.
Therefore, the correct answer is 4 ( x + 9 ) = 3 x − 8 − 5 .
Explanation
Understanding the Subtraction Property of Equality We are given the equation 4 ( x + 9 ) + 5 = 3 x − 8 and asked to identify which of the provided options demonstrates the subtraction property of equality. The subtraction property of equality states that if a = b , then a − c = b − c for any a , b , c . In simpler terms, we can subtract the same value from both sides of an equation without changing its solution.
Analyzing the Options Let's examine each option:
4 x + 36 + 5 = 3 x − 8 : This equation is obtained by distributing the 4 on the left side of the original equation: 4 ( x + 9 ) = 4 x + 36 . This demonstrates the distributive property, not the subtraction property.
x = − 49 : This is the solution to the equation. While correct, it doesn't directly show the subtraction property.
4 ( x + 9 ) − 5 + 8 = 3 x : This equation is not obtained by simply subtracting a constant from both sides of the original equation. It appears to involve multiple operations.
4 ( x + 9 ) = 3 x − 8 − 5 : This equation is obtained by subtracting 5 from both sides of the original equation: 4 ( x + 9 ) + 5 − 5 = 3 x − 8 − 5 , which simplifies to 4 ( x + 9 ) = 3 x − 13 or 4 ( x + 9 ) = 3 x − 8 − 5 . This clearly demonstrates the subtraction property of equality.
Identifying the Correct Option Therefore, the equation that demonstrates the subtraction property of equality is 4 ( x + 9 ) = 3 x − 8 − 5 .
Examples
The subtraction property of equality is a fundamental concept in algebra. It's used in many real-world scenarios, such as balancing a budget. Imagine you have a certain amount of money and you want to track your spending. If you subtract the same amount from both your income and expenses, the balance remains the same. For example, if you earn $100 and spend $30, your balance is $70. If you then receive a $20 bonus and spend an additional $20, your income is now $120 and your expenses are $50, but your balance still remains $70. This illustrates how subtracting the same amount from both sides of an equation (or in this case, a financial balance) maintains the equality.
