What is the point of inte \( \begin{array}{l}y=3 x-7 \\ 2 x+y=13\end{array} \)

To find the point of intersection of the lines represented by the equations \(y = 3x - 7\) and \(2x + y = 13\), we can substitute the expression for \(y\) from the first equation into the second equation. Substituting gives: \[ 2x + (3x - 7) = 13 \] This simplifies to: \[ 5x - 7 = 13 \] Adding 7 to both sides gives: \[ 5x = 20 \] Dividing by 5, we find: \[ x = 4 \] Now, plug the value of \(x\) back into the first equation to find \(y\): \[ y = 3(4) - 7 = 12 - 7 = 5 \] So, the point of intersection is \((4, 5)\). When looking at points of intersection in real-life systems, such as in economics or engineering, these points can represent optimal solutions, like maximizing profit or minimizing cost. Finding where two systems meet is crucial for making informed decisions! If you're solving systems of equations, one of the most common mistakes is forgetting to arrange equations in the same form. Keeping equations in standard form helps prevent errors in substitution or elimination processes. Always double-check your arithmetic too—small mistakes can lead to incorrect intersection points!