Which expressions are equivalent?
$\frac{1}{5} k-\frac{2}{3} j$ and $-\frac{2}{3} j+\frac{1}{5} k$
$\frac{1}{5} k-\frac{2}{3} j$ and $-\frac{1}{5} k+\frac{2}{3} j$
$\frac{1}{5} k-\frac{2}{3} j$ and $\frac{1}{5} j-\frac{2}{3} k$
$\frac{1}{5} k-\frac{2}{3} j$ and $\frac{2}{3} j-\frac{1}{5} k$
Answer (1)
The expression 5 1 k − 3 2 j is equivalent to − 3 2 j + 5 1 k due to the commutative property of addition. The other expressions are not equivalent because they have different signs or coefficients. Therefore, the equivalent expression is: − 3 2 j + 5 1 k .
Explanation
Understanding the Problem We are given the expression 5 1 k − 3 2 j . We need to determine which of the other given expressions are equivalent to this expression.
Plan of Action Let's analyze each expression to see if it is equivalent to the given expression.
Analyzing Expression 1 Expression 1: − 3 2 j + 5 1 k . This is equivalent to 5 1 k − 3 2 j due to the commutative property of addition.
Analyzing Expression 2 Expression 2: − 5 1 k + 3 2 j . This is not equivalent to 5 1 k − 3 2 j because the signs of both terms are different.
Analyzing Expression 3 Expression 3: 5 1 j − 3 2 k . This is not equivalent to 5 1 k − 3 2 j because the coefficients are switched between j and k.
Analyzing Expression 4 Expression 4: 3 2 j − 5 1 k . This is not equivalent to 5 1 k − 3 2 j because the signs of both terms are different.
Conclusion Therefore, only Expression 1 is equivalent to the given expression.
Examples
Understanding equivalent expressions is crucial in algebra, just like understanding that rearranging your grocery list doesn't change what you need to buy. For instance, if you're calculating the area of a garden bed and one part involves 5 1 of the total length minus 3 2 of the width, knowing that − 3 2 of the width plus 5 1 of the total length gives the same result helps simplify your calculations. This concept is fundamental in simplifying equations and solving real-world problems efficiently.
