Sure, let's solve the given problem step by step.
We start with the expression:
[tex]\[ x^2 + xy + y^2 \][/tex]
We need to substitute [tex]\(x\)[/tex] with [tex]\(x - y\)[/tex] in this expression. This means wherever we see [tex]\(x\)[/tex] in the expression, we replace it with [tex]\(x - y\)[/tex].
So substitute [tex]\(x\)[/tex] with [tex]\(x - y\)[/tex] in [tex]\(x^2 + xy + y^2\)[/tex]:
[tex]\[ (x - y)^2 + (x - y)y + y^2 \][/tex]
Now, let’s expand and simplify each term.
1. Expand [tex]\( (x - y)^2 \)[/tex]:
[tex]\[ (x - y)^2 = x^2 - 2xy + y^2 \][/tex]
2. Expand [tex]\( (x - y)y \)[/tex]:
[tex]\[ (x - y)y = xy - y^2 \][/tex]
Therefore, we now have:
[tex]\[ x^2 - 2xy + y^2 + xy - y^2 \][/tex]
Next, let's combine like terms. Notice that [tex]\( y^2 \)[/tex] and [tex]\( -y^2 \)[/tex] cancel each other out:
[tex]\[ x^2 - 2xy + xy + y^2 - y^2 \][/tex]
[tex]\[ x^2 - xy \][/tex]
So, the simplified expression after substituting [tex]\( x - y \)[/tex] into [tex]\( x^2 + xy + y^2 \)[/tex] is:
[tex]\[ x^2 - xy + y^2 \][/tex]
Thus, after substitution and simplification, we obtain the result:
[tex]\[ x^2 - xy + y^2 \][/tex]
