Consider the given system of equations where \( m \) and \( q \) are the slopes of the lines representer \( \left\{\begin{array}{l}y=m x+2 \\ y=q x+8\end{array}\right. \) The solution to the system of equations is the ordered pair \( (2,3) \)
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To find the slopes \( m \) and \( q \), we can substitute the ordered pair \( (2, 3) \) into both equations. For the first equation \( y = mx + 2 \), substituting \( x = 2 \) and \( y = 3 \) gives us: \[ 3 = m \cdot 2 + 2 \implies m \cdot 2 = 3 - 2 \implies m = \frac{1}{2}. \] Now for the second equation \( y = qx + 8 \): \[ 3 = q \cdot 2 + 8 \implies q \cdot 2 = 3 - 8 \implies q \cdot 2 = -5 \implies q = -\frac{5}{2}. \] So the slopes are \( m = \frac{1}{2} \) and \( q = -\frac{5}{2} \). Now, that’s a perfect example of how these lines interact—one slopes gently upwards while the other dives down sharply! If you enjoy solving systems of equations, try plotting them! It's quite fascinating to see how the lines approach each other. You can also play with the slopes: what happens when \( m \) and \( q \) are both positive or both negative? It's a fun way to visualize their relationships!
