To solve the system of linear equations:

1. $-x + 4y = -2$
2. $3x + 2y = 1$

We can use either substitution or elimination. Let’s solve it using elimination.

Step 1: Align the equations for elimination

The system of equations is: -x + 4y = -2 \tag{1} 3x + 2y = 1 \tag{2}

Step 2: Eliminate one variable

To eliminate $x$, multiply the first equation by 3 and the second equation by 1: 3(-x + 4y) = 3(-2) \implies -3x + 12y = -6 \tag{3} 3x + 2y = 1 \tag{4}

Now add equations (3) and (4): $(-3x + 12y) + (3x + 2y) = -6 + 1$ $14y = -5$

Solve for $y$: $y = -\frac{5}{14}$

Step 3: Solve for $x$

Substitute $y = -\frac{5}{14}$ into one of the original equations. Let’s use $-x + 4y = -2$: $-x + 4\left(-\frac{5}{14}\right) = -2$ $-x - \frac{20}{14} = -2$ $-x - \frac{10}{7} = -2$ Add $-\frac{10}{7}$ to both sides: $-x = -2 + \frac{10}{7}$ $-x = -\frac{14}{7} + \frac{10}{7}$ $-x = -\frac{4}{7}$ $x = \frac{4}{7}$

Final Solution:

$x = \frac{4}{7}, \; y = -\frac{5}{14}$

Would you like a detailed verification or additional steps explained?

Related Questions:

1. How would this system of equations be solved using substitution instead of elimination?
2. What is the geometric interpretation of solving a system of two linear equations?
3. Can you graphically verify the solution of this system?
4. How can this method be extended to three-variable systems?
5. What is the determinant's role in solving linear systems using matrices?

Tip: Always double-check your substitution or elimination steps to avoid arithmetic mistakes!