To determine if the point $\left(\frac{3}{2}, -1\right)$ lies on both lines of the given system of equations, we substitute $x = \frac{3}{2}$ and $y = -1$ into each equation.

First Equation:

$x + 2y + 5 = 0$ Substitute $x = \frac{3}{2}$ and $y = -1$: $\frac{3}{2} + 2(-1) + 5 = \frac{3}{2} - 2 + 5 = \frac{3}{2} + 3 = \frac{9}{2} \neq 0$ Thus, the point does not satisfy the first equation.

Since the point fails the first equation, it cannot lie on both lines of the system.

Answer:

No, the point $\left(\frac{3}{2}, -1\right)$ does not lie on both lines.

Would you like me to explore the process further or analyze specific parts of this?

Here are 5 related questions for exploration:

1. How do you verify solutions for systems of equations graphically?
2. What is the meaning of a point lying on both lines in a system of equations?
3. How do we find the intersection of two lines algebraically?
4. Can this system of equations have a solution if this point fails?
5. What are other methods to solve a system of linear equations besides substitution?

Tip: Always double-check calculations when substituting points into equations to ensure accuracy.