What is the solution to this system of linear equation \( 2 x+y=1 \) \( 3 x-y=-6 \) \( (-1,3) \) \( (1,-1) \) \( (2,3) \) \( (5,0) \)

To solve the system of equations given by \( 2x + y = 1 \) and \( 3x - y = -6 \), we can use substitution or elimination. If we re-arrange the first equation \( y = 1 - 2x \) and substitute into the second equation, we get \( 3x - (1 - 2x) = -6 \). Simplifying this leads to \( 3x - 1 + 2x = -6 \), which simplifies to \( 5x - 1 = -6 \) and then to \( 5x = -5 \), giving \( x = -1 \). Substituting \( x = -1 \) back into the first equation yields \( y = 1 - 2(-1) = 3 \). So the solution is \( (-1, 3) \). Now, let’s think about how these equations represent lines on a graph. Each equation corresponds to a straight line, and the solution we found, \( (-1, 3) \), represents the point where these two lines intersect. Visualizing these equations can make it clear why this point is the answer; finding where two paths cross is often a great way to understand relationships in math!