The image shows a system of two equations that need to be solved graphically. Here's the system of inequalities provided:

$y \geq x - 3$ $y \leq -\frac{1}{2}x - 1$

We are asked to find the coordinates of a point in the solution set.

Step-by-step solution:

1. Graph the first inequality $y \geq x - 3$:

   • The line equation is $y = x - 3$.
   • The slope is 1 and the y-intercept is -3.
   • The inequality $y \geq x - 3$ means we shade the region above the line.

2. Graph the second inequality $y \leq -\frac{1}{2}x - 1$:

   • The line equation is $y = -\frac{1}{2}x - 1$.
   • The slope is $-\frac{1}{2}$ and the y-intercept is -1.
   • The inequality $y \leq -\frac{1}{2}x - 1$ means we shade the region below the line.

3. Intersection of the two regions:

   • The solution set is the region where both inequalities overlap.
   • To find this, we can solve the system algebraically to find the boundary point where the two lines intersect.

Solving the system algebraically:

Set the equations of the two lines equal to each other: $x - 3 = -\frac{1}{2}x - 1$ Multiply the entire equation by 2 to eliminate the fraction: $2(x - 3) = -(x + 2)$ This simplifies to: $2x - 6 = -x - 2$ Add $x$ to both sides: $3x - 6 = -2$ Add 6 to both sides: $3x = 4$ Divide by 3: $x = \frac{4}{3}$

Substitute $x = \frac{4}{3}$ back into one of the original equations to find $y$. Let's use $y = x - 3$: $y = \frac{4}{3} - 3 = \frac{4}{3} - \frac{9}{3} = -\frac{5}{3}$

Thus, the lines intersect at the point $\left(\frac{4}{3}, -\frac{5}{3}\right)$.

This is the boundary point of the solution set, but the actual solution includes all points in the shaded region above and below the lines.

Would you like further details or have any questions?

Related questions:

1. How do you solve systems of linear equations algebraically?
2. What is the significance of the slope and y-intercept in graphing lines?
3. How do you determine the region to shade when graphing inequalities?
4. How can we graph systems of inequalities using software tools like Desmos?
5. What other methods can be used to solve systems of inequalities besides graphing?

Tip:

When graphing inequalities, always remember that a solid line indicates the inequality includes the boundary (≥ or ≤), while a dashed line indicates it does not ( > or < ).