Write three expressions that are equivalent to 3x.
The Correct answer and Explanation is:
To find three expressions that are equivalent to 3x3x3x, we can manipulate the expression using algebraic principles. Here are three equivalent expressions:
- x+x+xx + x + xx+x+x
- 6⋅x26 \cdot \frac{x}{2}6⋅2x
- 9×3\frac{9x}{3}39x
Explanation
- Breaking Down the Expression: The expression 3x3x3x can be understood as adding the variable xxx three times. Thus, x+x+xx + x + xx+x+x represents three instances of xxx, which simplifies to 3x3x3x. This is a fundamental concept in algebra where we can express multiplication as repeated addition.
- Using Multiplication with a Coefficient: In the second expression, 6⋅x26 \cdot \frac{x}{2}6⋅2x, we apply the property of multiplication that states multiplying a number by a fraction is equivalent to multiplying by the numerator and then dividing by the denominator. Here, 6⋅x26 \cdot \frac{x}{2}6⋅2x can be simplified as follows:
- First, calculate x2\frac{x}{2}2x, which represents half of xxx.
- Then, multiply that by 6, giving us 3x3x3x because 6⋅x2=3×6 \cdot \frac{x}{2} = 3×6⋅2x=3x. This manipulation shows how we can express the original expression using different numbers and operations while retaining its value.
- Using Division and Multiplication: The third expression, 9×3\frac{9x}{3}39x, utilizes division to create an equivalent form. Here’s how it works:
- The number 9 is divisible by 3, which simplifies to 3. Thus, when we divide 9x9x9x by 3, we essentially revert back to 3x3x3x. This expression demonstrates how multiplication and division can interchangeably represent the same quantity, showing the flexibility of algebraic expressions.
In conclusion, these equivalent expressions illustrate the principle of equivalence in algebra, where different arrangements and combinations of numbers and variables can yield the same result, allowing for a versatile approach to algebraic manipulation.