To solve the system of equations using substitution, the equations are:

1. $5x - 2y = 8$
2. $3x + 4y = 10$

Step 1: Solve one equation for one variable

From the first equation $5x - 2y = 8$, solve for $y$ in terms of $x$:

$-2y = 8 - 5x \quad \Rightarrow \quad y = \frac{5x - 8}{2}$

Step 2: Substitute $y = \frac{5x - 8}{2}$ into the second equation

Substitute this expression for $y$ into $3x + 4y = 10$:

$3x + 4\left(\frac{5x - 8}{2}\right) = 10$

Simplify:

$3x + 2(5x - 8) = 10 \quad \Rightarrow \quad 3x + 10x - 16 = 10$

$13x - 16 = 10 \quad \Rightarrow \quad 13x = 26$

$x = 2$

Step 3: Solve for $y$

Substitute $x = 2$ back into $y = \frac{5x - 8}{2}$:

$y = \frac{5(2) - 8}{2} = \frac{10 - 8}{2} = \frac{2}{2} = 1$

Final Solution:

The solution to the system is: $x = 2, \, y = 1$

Let me know if you want further details or have any questions!

Related Questions:

1. How would the solution change if solved using elimination instead of substitution?
2. Can you verify the solution by substituting into both equations?
3. What does the graph of this system look like?
4. What is the determinant of the coefficient matrix for this system of equations?
5. How would the solution be affected if one equation were modified?

Tip:

Always verify your solution by substituting it back into the original equations to confirm its accuracy!