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Let's solve each of the parts of the problem given.

### Problem Statement
Given the linear equation:
\[ 2y - 3x = 5 \]

1. **Find another solution for the equation if \((-1, 1)\) is already a solution.**

We need to find another \((x, y)\) pair that satisfies this equation. Let's choose \(x = 0\) (a simple value) and substitute it into the equation:
\[
2y - 3(0) = 5 \implies 2y = 5 \implies y = \frac{5}{2}
\]
So, another solution is \((0, \frac{5}{2})\).

2. **Check if the points \((-1, 1)\) and \((4, 1)\) are solutions to the inequality \(2y - 3x < 5\).**

Let's substitute each point into the inequality:

- For \((-1, 1)\):
  \[
  2(1) - 3(-1) = 2 + 3 = 5
  \]
  Since \(5 \not< 5\), \((-1, 1)\) is **not a solution** to the inequality.

- For \((4, 1)\):
  \[
  2(1) - 3(4) = 2 - 12 = -10
  \]
  Since \(-10 < 5\), \((4, 1)\) **is a solution** to the inequality.

3. **Explain how to use the answers to the previous questions to graph the solution set to the inequality.**

To graph the solution to the inequality \(2y - 3x < 5\):
- First, graph the line \(2y - 3x = 5\) by plotting the points \((-1, 1)\) and \((0, \frac{5}{2})\) and drawing a straight line through them.
- Since the inequality is strict (\(<\)), the line should be dashed to indicate that points on the line are not included in the solution.
- Test a point not on the line (like the origin \((0,0)\)). Substituting it into the inequality:
  \[
  2(0) - 3(0) = 0 < 5
  \]
  Since the origin satisfies the inequality, shade the region of the graph that contains the origin.

Would you like more details on this or have any other questions? 

Here are some follow-up questions that might be relevant:
1. How do you determine if a point lies on the boundary line of an inequality?
2. What would change in the graph if the inequality were \(2y - 3x \leq 5\) instead?
3. How do you find the x- and y-intercepts of a linear equation?
4. How can you tell if a system of linear equations has no solution, one solution, or infinitely many solutions?
5. How does the slope of a line relate to its graphical representation?

**Tip:** Always check the inequality with a test point after shading to confirm you have the correct region.

Consider the linear equation 2y - 3x = 5. 1. The pair (-1, 1) is a solution to the equation. Find another (x, y) pair that is a solution to the equation. 2. Are (-1, 1) and (4, 1) solutions to the inequality 2y - 3x < 5? Explain how you know. 3. Explain how to use the answers to the previous questions to graph the solution set to the inequality.

Learn how to solve and graph solutions for the linear equation 2y - 3x = 5 and the inequality 2y - 3x < 5. This step-by-step explanation covers algebraic substitution, testing points for inequalities, and graphing techniques. Suitable for students in Grades 8-10.

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