To solve the problem of finding the expression that must be subtracted from [tex]\( x^2 + 5y^2 - 3xy \)[/tex] to get [tex]\( 2x^2 - y^2 + 4xy \)[/tex], follow these steps:
1. Define the initial expressions:
Let the initial expression be [tex]\( A = x^2 + 5y^2 - 3xy \)[/tex].
Let the resulting expression be [tex]\( B = 2x^2 - y^2 + 4xy \)[/tex].
2. Understand the problem:
We need to find an unknown expression [tex]\( C \)[/tex] such that:
[tex]\[
A - C = B
\][/tex]
3. Rearrange the equation to isolate [tex]\( C \)[/tex]:
To find [tex]\( C \)[/tex], rearrange the equation:
[tex]\[
C = A - B
\][/tex]
4. Substitute the given expressions for [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
Substitute [tex]\( A = x^2 + 5y^2 - 3xy \)[/tex] and [tex]\( B = 2x^2 - y^2 + 4xy \)[/tex]:
[tex]\[
C = (x^2 + 5y^2 - 3xy) - (2x^2 - y^2 + 4xy)
\][/tex]
5. Distribute the subtraction:
Simplify by distributing the negative sign across the second expression:
[tex]\[
C = x^2 + 5y^2 - 3xy - 2x^2 + y^2 - 4xy
\][/tex]
6. Combine like terms:
Combine the [tex]\( x^2 \)[/tex] terms: [tex]\( x^2 - 2x^2 = -x^2 \)[/tex].
Combine the [tex]\( y^2 \)[/tex] terms: [tex]\( 5y^2 + y^2 = 6y^2 \)[/tex].
Combine the [tex]\( xy \)[/tex] terms: [tex]\( -3xy - 4xy = -7xy \)[/tex].
7. Write the final expression for [tex]\( C \)[/tex]:
The expression that must be subtracted is:
[tex]\[
C = -x^2 + 6y^2 - 7xy
\][/tex]
Therefore, the expression that must be subtracted from [tex]\( x^2 + 5y^2 - 3xy \)[/tex] to get [tex]\( 2x^2 - y^2 + 4xy \)[/tex] is:
[tex]\[
-x^2 + 6y^2 - 7xy
\][/tex]
