To find the corners of the solution region, we solve the system of inequalities by determining the points of intersection between the boundary lines. Here are the steps:

Step 1: Solve for the intersection of the two lines

The boundary equations of the inequalities are:

1. $4x + 2y = 8$
2. $5x + y = 7$

Substitution or elimination method:

From equation (2): $y = 7 - 5x$.

Substitute $y = 7 - 5x$ into equation (1): $4x + 2(7 - 5x) = 8$ $4x + 14 - 10x = 8$ $-6x + 14 = 8$ $-6x = -6$ $x = 1$

Now substitute $x = 1$ into $y = 7 - 5x$: $y = 7 - 5(1) = 2$

So, the intersection point is $(1, 2)$.

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Step 2: Find other intersection points

• Line $4x + 2y = 8$:

  • When $x = 0$, $4(0) + 2y = 8$ → $y = 4$. Point: $(0, 4)$.
  • When $y = 0$, $4x + 2(0) = 8$ → $x = 2$. Point: $(2, 0)$.

• Line $5x + y = 7$:

  • When $x = 0$, $5(0) + y = 7$ → $y = 7$. Point: $(0, 7)$.
  • When $y = 0$, $5x + 0 = 7$ → $x = 7/5 = 1.4$. Point: $(1.4, 0)$.

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Step 3: Identify the corners of the solution region

The shaded region is the area above both lines and in the first quadrant. This means we consider only feasible points in the first quadrant and above both boundary lines.

The corners of the solution region are:

1. Intersection of $4x + 2y = 8$ and $5x + y = 7$: $(1, 2)$.
2. Intersection of $4x + 2y = 8$ with the y-axis: $(0, 4)$.
3. Intersection of $5x + y = 7$ with the y-axis: $(0, 7)$.

Thus, the corners of the solution region are: $(1, 2), (0, 4), (0, 7)$

Let me know if you'd like a more detailed explanation or help visualizing this solution.

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Relative Questions:

1. How do you determine whether a point lies in the solution region of a system of inequalities?
2. Why is the solution region bounded by the first quadrant in this problem?
3. What method is best for solving systems of linear equations graphically versus algebraically?
4. How do you test if a corner point satisfies all inequalities in a system?
5. What are the practical applications of finding solution regions in inequalities?

Tip:

When solving graphically, always double-check boundary intersections and confirm if the shaded region satisfies the inequalities.