Let's break down the given options to determine which algebraic expression Oneta could have written based on the criteria:
1. The expression must have three terms.
2. The [tex]\( y \)[/tex]-term must have a coefficient of [tex]\(-3\)[/tex].
3. The [tex]\( x \)[/tex]-term must have a coefficient of 1.
4. The expression should not have a constant term.
Let's evaluate each given option against this criteria:
1. Option 1: [tex]\( x - y^2 - 3y \)[/tex]
- This expression has three terms: [tex]\( x \)[/tex], [tex]\( -y^2 \)[/tex], and [tex]\( -3y \)[/tex].
- The [tex]\( y \)[/tex]-term here is [tex]\( -3y \)[/tex].
- The [tex]\( x \)[/tex]-term here is [tex]\( x \)[/tex], which implies a coefficient of 1.
- This option does not have a constant term.
- This fits all the given criteria.
2. Option 2: [tex]\( x - 3y + 6 \)[/tex]
- This expression has three terms: [tex]\( x \)[/tex], [tex]\( -3y \)[/tex], and [tex]\( 6 \)[/tex].
- The [tex]\( y \)[/tex]-term here is [tex]\( -3y \)[/tex].
- The [tex]\( x \)[/tex]-term here is [tex]\( x \)[/tex], which implies a coefficient of 1.
- This option has a constant term ([tex]\( 6 \)[/tex]), which does not fit the criteria.
3. Option 3: [tex]\( x + 3y^2 + 3y \)[/tex]
- This expression has three terms: [tex]\( x \)[/tex], [tex]\( 3y^2 \)[/tex], and [tex]\( 3y \)[/tex].
- The [tex]\( y \)[/tex]-term here is [tex]\( 3y \)[/tex], which has a coefficient of [tex]\( 3 \)[/tex] (not [tex]\( -3 \)[/tex]).
- The [tex]\( x \)[/tex]-term here is [tex]\( x \)[/tex], which implies a coefficient of 1.
- This option does not have a constant term.
- This does not fit the criterion for the [tex]\( y \)[/tex]-term coefficient.
4. Option 4: [tex]\( x + 3y + 7 \)[/tex]
- This expression has three terms: [tex]\( x \)[/tex], [tex]\( 3y \)[/tex], and [tex]\( 7 \)[/tex].
- The [tex]\( y \)[/tex]-term here is [tex]\( 3y \)[/tex], which has a coefficient of [tex]\( 3 \)[/tex] (not [tex]\( -3 \)[/tex]).
- The [tex]\( x \)[/tex]-term here is [tex]\( x \)[/tex], which implies a coefficient of 1.
- This option has a constant term ([tex]\( 7 \)[/tex]), which does not fit the criteria.
Putting this all together, the only option that meets all the specified requirements is:
[tex]\[ \boxed{x - y^2 - 3y} \][/tex]
