We are tasked with solving the system of linear equations using substitution:

1. $2x + 5y = 11$
2. $4x + 3y = 1$

Step 1: Solve one equation for one variable

From the first equation ($2x + 5y = 11$), solve for $x$ in terms of $y$:
$2x = 11 - 5y \quad \Rightarrow \quad x = \frac{11 - 5y}{2}.$

Step 2: Substitute $x = \frac{11 - 5y}{2}$ into the second equation

Substitute $x$ into the second equation ($4x + 3y = 1$): $4\left(\frac{11 - 5y}{2}\right) + 3y = 1.$

Simplify the terms: $2(11 - 5y) + 3y = 1 \quad \Rightarrow \quad 22 - 10y + 3y = 1 \quad \Rightarrow \quad 22 - 7y = 1.$

Solve for $y$: $-7y = 1 - 22 \quad \Rightarrow \quad -7y = -21 \quad \Rightarrow \quad y = 3.$

Step 3: Substitute $y = 3$ back into $x = \frac{11 - 5y}{2}$

Now substitute $y = 3$ into $x = \frac{11 - 5y}{2}$: $x = \frac{11 - 5(3)}{2} = \frac{11 - 15}{2} = \frac{-4}{2} = -2.$

Final Solution:

The solution to the system of equations is: $x = -2, \, y = 3.$

Would you like further clarification or details on any of the steps? Let me know!

Related Questions:

1. What is substitution, and why does it work for solving systems of equations?
2. How can you solve the same system of equations using elimination instead?
3. How can you verify the solution $x = -2, y = 3$ by substitution back into both equations?
4. What happens if the system of equations has no solution or infinite solutions?
5. Can substitution be used for non-linear systems of equations?

Tip:

When solving by substitution, always simplify the expressions as much as possible before substituting back to avoid errors.