To determine which algebraic expression Oneta could have written given the constraints:
1. The [tex]$y$[/tex]-term has a coefficient of -3.
2. The [tex]$x$[/tex]-term has a coefficient of 1.
3. The expression does not have a constant term.
Let's analyze the given options one by one:
1. \( x - y^2 - 3y \):
- The [tex]$x$[/tex]-term has a coefficient of 1, which matches the given condition.
- The [tex]$y$[/tex]-term here is \(-3y\), which has a coefficient of -3, also matching the given condition.
- There is no constant term in the expression.
- Therefore, this expression fits all the given criteria.
2. \( x - 3y + 6 \):
- The [tex]$x$[/tex]-term has a coefficient of 1, which matches the condition.
- The [tex]$y$[/tex]-term has a coefficient of -3, which also matches the condition.
- However, the expression includes a constant term of 6, which does not fit the condition of having no constant term.
- Therefore, this expression does not meet all the criteria.
3. \( x + 3y^2 + 3y \):
- The [tex]$x$[/tex]-term has a coefficient of 1, which matches the condition.
- The [tex]$y$[/tex]-term has a coefficient of 3, not -3.
- There is no constant term in the expression.
- Therefore, this expression does not meet the criteria regarding the [tex]$y$[/tex]-term's coefficient.
4. \( x + 3y + 7 \):
- The [tex]$x$[/tex]-term has a coefficient of 1, which matches the condition.
- The [tex]$y$[/tex]-term has a coefficient of 3, not -3.
- The expression includes a constant term of 7, which does not fit the condition of having no constant term.
- Therefore, this expression does not meet multiple criteria.
From the analysis above, the only expression that matches all the conditions given is:
[tex]\[ x - y^2 - 3y \][/tex]
So, Oneta could have written the expression [tex]\( x - y^2 - 3y \)[/tex].
