Rahul solved the equation $2\left(x-\frac{1}{8}\right)-\frac{3}{5} x=\frac{55}{4}$. In which step did he use the addition property of equality?
Rahul's Solution
| Steps | Resulting equations |
| :---- | :------------------ |
| 1 | $2 x-\frac{1}{4}-\frac{3}{5} x=\frac{55}{4}$ |
| 2 | $\frac{7}{5} x-\frac{1}{4}=\frac{55}{4}$ |
| 3 | $\frac{7}{5} x=\frac{56}{4}$ |
| 4 | $x=10$ |
A. Step 1
B. Step 2
C. Step 3
Answer (1)
Step 1 uses the distributive property.
Step 2 combines like terms.
Step 3 uses the addition property of equality.
Step 4 uses the multiplication property of equality.
The addition property of equality was used in Step 3 .
Explanation
Understanding the Problem We are given Rahul's solution to the equation 2 ( x − 8 1 ) − 5 3 x = 4 55 and we need to determine in which step he used the addition property of equality. The addition property of equality states that if a = b , then a + c = b + c for any c . In other words, we can add the same value to both sides of an equation without changing the equality.
Identifying the Use of Addition Property of Equality Let's examine each step in Rahul's solution to see where the addition property of equality was used.
Step 1: 2 x − 4 1 − 5 3 x = 4 55 . This step is obtained by distributing the 2 in the original equation: 2 ( x − 8 1 ) − 5 3 x = 2 x − 8 2 − 5 3 x = 2 x − 4 1 − 5 3 x . So, Step 1 uses the distributive property.
Step 2: 5 7 x − 4 1 = 4 55 . This step is obtained by combining like terms in Step 1: 2 x − 5 3 x − 4 1 = 5 10 x − 5 3 x − 4 1 = 5 7 x − 4 1 . So, Step 2 combines like terms.
Step 3: 5 7 x = 4 56 . Comparing Step 2 and Step 3, we see that 4 1 was added to both sides of the equation in Step 2: 5 7 x − 4 1 + 4 1 = 4 55 + 4 1 , which simplifies to 5 7 x = 4 56 . This is the addition property of equality.
Step 4: x = 10 . This step is obtained by multiplying both sides of the equation in Step 3 by 7 5 : 7 5 ⋅ 5 7 x = 7 5 ⋅ 4 56 , which simplifies to x = 7 ⋅ 4 5 ⋅ 56 = 4 5 ⋅ 8 = 5 ⋅ 2 = 10 . This step uses the multiplication property of equality.
Conclusion Therefore, Rahul used the addition property of equality in Step 3.
Examples
The addition property of equality is a fundamental concept in algebra and is used in many real-world applications. For example, if you are trying to balance a budget and you want to add a certain amount to both your income and expenses, you are using the addition property of equality. Similarly, if you are trying to solve a physics problem and you need to add a certain force to both sides of an equation, you are using the addition property of equality. The addition property of equality ensures that the equation remains balanced and that the solution remains valid.
