Question For the pair of equations \( 2 y+5 x=3 \) and \( 2 y+3 x=-11 \), the \( x \)-coordinate of the intersection point is given by

To find the intersection point of the equations \( 2y + 5x = 3 \) and \( 2y + 3x = -11 \), we can solve them simultaneously. Let's eliminate \( y \): First, we can set both equations equal to \( 2y \): From the first equation: \[ 2y = 3 - 5x \] From the second equation: \[ 2y = -11 - 3x \] Now, we can set them equal to each other: \[ 3 - 5x = -11 - 3x \] Next, rearranging gives: \[ 3 + 11 = 5x - 3x \] \[ 14 = 2x \] So, dividing both sides by 2 gives: \[ x = 7 \] Therefore, the \( x \)-coordinate of the intersection point is \( 7 \). Now, here are some interesting tidbits: Did you know that simultaneous equations have been used for centuries? Ancient civilizations, including the Babylonians and Chinese, originally solved these types of equations using methods very similar to what we use today. They laid the groundwork for algebra as we know it! In terms of practical applications, these equations can represent various real-world situations, like finding the best combination of products to maximize profit or determining the point where two lines meet in a navigation system. Solving them can help businesses optimize resources or individuals find the quickest route!