To find an equivalent equation that can replace [tex]\(3x + 5y = 59\)[/tex] in the given system without changing the solution, we need to consider the process of elimination and linear combinations.
The original system of equations is:
1. [tex]\(3x + 5y = 59\)[/tex]
2. [tex]\(2x - y = -4\)[/tex]
We want to transform the second equation in such a way that it can replace the first equation but will still maintain the solution. Here's how we can do it:
1. Multiply the second equation by a factor that will help eliminate one of the terms when added to or subtracted from another equation. For instance, by multiplying the second equation by 5, we get:
[tex]\[
5 \cdot (2x - y) = 5 \cdot (-4)
\][/tex]
which simplifies to:
[tex]\[
10x - 5y = -20
\][/tex]
2. Now we look at our original first equation:
[tex]\[
3x + 5y = 59
\][/tex]
3. We can now consider adding these two equations together to see if we can eliminate the [tex]\(y\)[/tex] term:
[tex]\[
3x + 5y = 59 \\
10x - 5y = -20
\][/tex]
Adding these equations:
[tex]\[
(3x + 10x) + (5y - 5y) = 59 + (-20)
\][/tex]
simplifies to:
[tex]\[
13x = 39
\][/tex]
Thus, adding these modified equations results in the simplified equation:
[tex]\[
13x = 39
\][/tex]
Therefore, the equivalent equation that can replace [tex]\(3x + 5y = 59\)[/tex] without changing the solution is:
[tex]\[
\boxed{13x = 39}
\][/tex]
