\( x+2 y=4 \) and \( 4 x+5 y=1 \)

To solve the system of equations, let's start with \( x + 2y = 4 \). You can express \( x \) in terms of \( y \): \( x = 4 - 2y \). Now, substitute this expression for \( x \) into the second equation \( 4x + 5y = 1 \): \[ 4(4 - 2y) + 5y = 1. \] This simplifies to: \[ 16 - 8y + 5y = 1, \] \[ 16 - 3y = 1. \] Now, isolate \( y \): \[ -3y = 1 - 16, \] \[ -3y = -15 \implies y = 5. \] Next, substitute \( y = 5 \) back into the expression for \( x \): \[ x = 4 - 2(5) = 4 - 10 = -6. \] Thus, the solution to the system of equations is \( x = -6 \) and \( y = 5 \). Both equations represent lines in a 2D plot, and their intersection point is the solution of this system. Graphically, \( x + 2y = 4 \) would show up as a straight line crossing the y-axis at 2 and the x-axis at 4, while \( 4x + 5y = 1 \) intersects both axes at significantly different points. Understanding the geometry helps visualize why there's a unique point where they intersect (if they do) in this case.