To solve the system of linear equations:

1. $2x + 3y = 14$
2. $3x + 2y = 8$

we can use either the substitution method or the elimination method. Here, I will use the elimination method to solve the system.

Step-by-Step Solution

Step 1: Align the equations

$2x + 3y = 14 \quad \text{(Equation 1)}$ $3x + 2y = 8 \quad \text{(Equation 2)}$

Step 2: Multiply the equations to align coefficients

To eliminate $x$, we can multiply Equation 1 by 3 and Equation 2 by 2:

$3(2x + 3y) = 3 \cdot 14 \implies 6x + 9y = 42$ $2(3x + 2y) = 2 \cdot 8 \implies 6x + 4y = 16$

Now we have the system:

$6x + 9y = 42 \quad \text{(Equation 3)}$ $6x + 4y = 16 \quad \text{(Equation 4)}$

Step 3: Subtract the equations to eliminate $x$

Subtract Equation 4 from Equation 3:

$(6x + 9y) - (6x + 4y) = 42 - 16$ $5y = 26$

Step 4: Solve for $y$

$y = \frac{26}{5} = 5.2$

Step 5: Substitute $y$ back into one of the original equations

Let's substitute $y = 5.2$ into Equation 1:

$2x + 3(5.2) = 14$ $2x + 15.6 = 14$ $2x = 14 - 15.6$ $2x = -1.6$ $x = \frac{-1.6}{2} = -0.8$

Solution:

$x = -0.8, \quad y = 5.2$

So, the solution to the system of equations is:

$(x, y) = (-0.8, 5.2)$

Would you like more details on any of these steps, or do you have any questions?

5 Related Questions:

1. How can we verify the solution to this system of equations?
2. What is the graphical interpretation of solving a system of linear equations?
3. How can we solve this system using the substitution method?
4. What are some real-world applications of solving systems of equations?
5. How can we solve systems of equations with more than two variables?

Tip: When solving systems of equations, it's helpful to check your solution by substituting the values back into the original equations to ensure they satisfy both equations.