Question

Show your work:
21. Solve this linear system by elimination:
5𝑥 + 3𝑦 = 11
2x + y = 4

Answer 1

To solve the linear system using the elimination method, we have the following equations:

1. \( 5x + 3y = 11 \) \quad (Equation 1)
2. \( 2x + y = 4 \) \quad (Equation 2)

First, let's align the equations for elimination. To eliminate \( y \), we need the coefficients of \( y \) in both equations to be the same.

Starting with Equation 2, we can multiply it by 3 to match the coefficient of \( y \) in Equation 1:

\[
3(2x + y) = 3(4)
\]

This results in:
\[
6x + 3y = 12 \quad (Equation 3)
\]

Now, we have the new system of equations:

1. \( 5x + 3y = 11 \) \quad (Equation 1)
2. \( 6x + 3y = 12 \) \quad (Equation 3)

Next, we subtract Equation 1 from Equation 3 to eliminate \( y \):

\[
(6x + 3y) - (5x + 3y) = 12 - 11
\]

This simplifies to:

\[
6x - 5x = 1
\]

Thus, we have:

\[
x = 1
\]

Now that we have the value of \( x \), we can substitute \( x = 1 \) back into Equation 2 to find \( y \):

\[
2(1) + y = 4
\]
\[
2 + y = 4
\]
\[
y = 4 - 2
\]
\[
y = 2
\]

Therefore, the solution to the system of equations is:

\[
(x, y) = (1, 2)
\]

**Final Answer: \( x = 1, y = 2 \)**.