To determine what should be subtracted from [tex]\(5x - 3y + 8\)[/tex] to obtain [tex]\(2x + y + 5\)[/tex], follow these step-by-step instructions:
1. Write the original expression and the target expression:
- The original expression is [tex]\(5x - 3y + 8\)[/tex].
- The target expression is [tex]\(2x + y + 5\)[/tex].
2. Set up an equation to find the unknown expression that, when subtracted from the original expression, results in the target expression:
- Let the unknown expression be [tex]\(ax + by + c\)[/tex].
3. Write the equation representing the problem:
[tex]\[
(5x - 3y + 8) - (ax + by + c) = 2x + y + 5
\][/tex]
4. Combine and simplify:
[tex]\[
5x - 3y + 8 - ax - by - c = 2x + y + 5
\][/tex]
5. Group like terms:
[tex]\[
(5x - ax) + (-3y - by) + (8 - c) = 2x + y + 5
\][/tex]
6. Compare coefficients on both sides of the equation:
- Coefficient of [tex]\(x\)[/tex]: [tex]\(5 - a = 2\)[/tex]
- Coefficient of [tex]\(y\)[/tex]: [tex]\(-3 - b = 1\)[/tex]
- Constant term: [tex]\(8 - c = 5\)[/tex]
7. Solve for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- For [tex]\(x\)[/tex]:
[tex]\[
5 - a = 2 \implies a = 3
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
-3 - b = 1 \implies b = -4
\][/tex]
- For the constant term:
[tex]\[
8 - c = 5 \implies c = 3
\][/tex]
8. Write the unknown expression:
The expression [tex]\(ax + by + c\)[/tex] is:
[tex]\[
3x - 4y + 3
\][/tex]
Hence, the expression [tex]\(3x - 4y + 3\)[/tex] should be subtracted from [tex]\(5x - 3y + 8\)[/tex] to get [tex]\(2x + y + 5\)[/tex].
