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

To solve the system of equations \( x + 2y = 4 \) and \( 4x + 5y = 1 \), you can use the substitution or elimination method. Let's rearrange the first equation for \( x \): \( x = 4 - 2y \). Now, substitute this expression for \( x \) into the second equation: \( 4(4 - 2y) + 5y = 1 \). Expanding gives you \( 16 - 8y + 5y = 1 \), which simplifies to \( 16 - 3y = 1 \). From here, solving for \( y \): \( -3y = 1 - 16 \) leads to \( -3y = -15 \), so \( y = 5 \). Now plug \( y = 5 \) back into \( x + 2(5) = 4 \) to find \( x \). Thus, \( x + 10 = 4 \) gives \( x = 4 - 10 = -6 \). So, the solution is \( (x, y) = (-6, 5) \). Visualizing systems of equations as lines on a graph can turn problem-solving into an exciting game! The point where the lines intersect represents the solution of the system. This perspective helps students grasp how multiple equations relate, transforming what could be a daunting task into a friendly challenge with creativity in mind! To enrich your understanding further, exploring linear programming can enhance your problem-solving skills. It involves optimizing a particular value while adhering to constraints expressed as linear equations. You can find dead-end routes when figuring out solutions without proper adjustments, but by delving into practical scenarios like maximizing profits or minimizing costs, you can use systems of equations to guide you towards real-world solutions!