Which facts could be applied to simplify this expression? Check all that apply.
[tex]5 x+3 y+(-x)+6 z[/tex]
A. To add like terms, add the coefficients, not the variables.
B. First add [tex]5 x+x[/tex]
C. The simplified expression is [tex]4 x+3 y+6 z[/tex].
D. Only combine terms which contain the same variable.
E. The simplified expression is [tex]5 x+3 y+6 z[/tex].
Answer (1)
Combine like terms by adding their coefficients: 5 x + ( − x ) = 4 x .
The simplified expression is 4 x + 3 y + 6 z .
Correct statements are: 'To add like terms, add the coefficients, not the variables', 'The simplified expression is 4 x + 3 y + 6 z ', and 'Only combine terms which contain the same variable'.
The final simplified expression is 4 x + 3 y + 6 z .
Explanation
Understanding the Expression We are given the expression 5 x + 3 y + ( − x ) + 6 z and asked to identify the correct statements about simplifying it.
Identifying Like Terms The expression contains terms with variables x , y , and z . We can only combine terms that have the same variable.
Combining Like Terms The like terms in the expression are 5 x and − x . To combine them, we add their coefficients: 5 + ( − 1 ) = 4 . So, 5 x + ( − x ) = 4 x .
Simplified Expression The simplified expression is 4 x + 3 y + 6 z .
Evaluating the Statements Now, let's evaluate the given statements:
"To add like terms, add the coefficients, not the variables." - This statement is correct.
"First add 5 x + x " - This statement is incorrect. We should add 5 x + ( − x ) or 5 x − x .
"The simplified expression is 4 x + 3 y + 6 z ." - This statement is correct.
"Only combine terms which contain the same variable." - This statement is correct.
"The simplified expression is 5 x + 3 y + 6 z ." - This statement is incorrect.
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics. For example, if you are calculating the perimeter of a shape with sides of length 5 x , 3 y , x , and 6 z , you would add these lengths together to get 5 x + 3 y + x + 6 z . Simplifying this expression to 6 x + 3 y + 6 z makes it easier to calculate the perimeter for different values of x , y , and z .
