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To solve the system of linear inequalities graphically and identify one solution, we will follow these steps:
1. Graph each inequality separately:
- Convert each inequality into an equation by replacing the inequality sign with an equality sign. This will help us identify the boundary lines.
2. Shade the feasible region for each inequality:
- Determine which side of the boundary line contains the solutions to the inequality by testing a point.
### Step-by-Step Solution:
#### Inequality 1: [tex]\( x - y \leq 3 \)[/tex]
1. Convert to equation: [tex]\( x - y = 3 \)[/tex]
2. Find boundary line points:
- When [tex]\( x = 0 \)[/tex], [tex]\( -y = 3 \)[/tex] ⟹ [tex]\( y = -3 \)[/tex] ⟹ Point (0, -3)
- When [tex]\( y = 0 \)[/tex], [tex]\( x = 3 \)[/tex] ⟹ Point (3, 0)
3. Graph the boundary line:
- Plot the points (0, -3) and (3, 0) on the graph and draw the line through these points.
4. Determine shading:
- Select a test point not on the line (e.g., (0, 0)).
- Test the point: [tex]\( 0 - 0 \leq 3 \)[/tex] ⟹ True.
- Since (0, 0) satisfies the inequality, shade the region below and to the left of the line.
#### Inequality 2: [tex]\( 2x + 3y \leq -9 \)[/tex]
1. Convert to equation: [tex]\( 2x + 3y = -9 \)[/tex]
2. Find boundary line points:
- When [tex]\( x = 0 \)[/tex], [tex]\( 3y = -9 \)[/tex] ⟹ [tex]\( y = -3 \)[/tex] ⟹ Point (0, -3)
- When [tex]\( y = 0 \)[/tex], [tex]\( 2x = -9 \)[/tex] ⟹ [tex]\( x = -4.5 \)[/tex] ⟹ Point (-4.5, 0)
3. Graph the boundary line:
- Plot the points (0, -3) and (-4.5, 0) on the graph and draw the line through these points.
4. Determine shading:
- Select a test point not on the line (e.g., (0, 0)).
- Test the point: [tex]\( 2(0) + 3(0) \leq -9 \)[/tex] ⟹ [tex]\( 0 \leq -9 \)[/tex] ⟹ False.
- Since (0, 0) does not satisfy the inequality, shade the region below and to the left of the line.
### Identifying the Solution:
Observe the shaded regions from both inequalities. The solutions to the system of inequalities are found in the region where the shaded areas overlap.
### One Solution:
One solution that lies within the region of overlap for both shaded areas is [tex]\( x = -6 \)[/tex] and [tex]\( y = 1 \)[/tex].
Let's verify this solution:
1. Substitute into the first inequality [tex]\( x - y \leq 3 \)[/tex]:
- [tex]\( -6 - 1 \leq 3 \)[/tex] ⟹ [tex]\( -7 \leq 3 \)[/tex] ⟹ True
2. Substitute into the second inequality [tex]\( 2x + 3y \leq -9 \)[/tex]:
- [tex]\( 2(-6) + 3(1) \leq -9 \)[/tex] ⟹ [tex]\( -12 + 3 \leq -9 \)[/tex] ⟹ -9 \leq -9 ⟹ True
Thus, [tex]\( (x, y) = (-6, 1) \)[/tex] satisfies both inequalities, confirming it as a valid solution.
1. Graph each inequality separately:
- Convert each inequality into an equation by replacing the inequality sign with an equality sign. This will help us identify the boundary lines.
2. Shade the feasible region for each inequality:
- Determine which side of the boundary line contains the solutions to the inequality by testing a point.
### Step-by-Step Solution:
#### Inequality 1: [tex]\( x - y \leq 3 \)[/tex]
1. Convert to equation: [tex]\( x - y = 3 \)[/tex]
2. Find boundary line points:
- When [tex]\( x = 0 \)[/tex], [tex]\( -y = 3 \)[/tex] ⟹ [tex]\( y = -3 \)[/tex] ⟹ Point (0, -3)
- When [tex]\( y = 0 \)[/tex], [tex]\( x = 3 \)[/tex] ⟹ Point (3, 0)
3. Graph the boundary line:
- Plot the points (0, -3) and (3, 0) on the graph and draw the line through these points.
4. Determine shading:
- Select a test point not on the line (e.g., (0, 0)).
- Test the point: [tex]\( 0 - 0 \leq 3 \)[/tex] ⟹ True.
- Since (0, 0) satisfies the inequality, shade the region below and to the left of the line.
#### Inequality 2: [tex]\( 2x + 3y \leq -9 \)[/tex]
1. Convert to equation: [tex]\( 2x + 3y = -9 \)[/tex]
2. Find boundary line points:
- When [tex]\( x = 0 \)[/tex], [tex]\( 3y = -9 \)[/tex] ⟹ [tex]\( y = -3 \)[/tex] ⟹ Point (0, -3)
- When [tex]\( y = 0 \)[/tex], [tex]\( 2x = -9 \)[/tex] ⟹ [tex]\( x = -4.5 \)[/tex] ⟹ Point (-4.5, 0)
3. Graph the boundary line:
- Plot the points (0, -3) and (-4.5, 0) on the graph and draw the line through these points.
4. Determine shading:
- Select a test point not on the line (e.g., (0, 0)).
- Test the point: [tex]\( 2(0) + 3(0) \leq -9 \)[/tex] ⟹ [tex]\( 0 \leq -9 \)[/tex] ⟹ False.
- Since (0, 0) does not satisfy the inequality, shade the region below and to the left of the line.
### Identifying the Solution:
Observe the shaded regions from both inequalities. The solutions to the system of inequalities are found in the region where the shaded areas overlap.
### One Solution:
One solution that lies within the region of overlap for both shaded areas is [tex]\( x = -6 \)[/tex] and [tex]\( y = 1 \)[/tex].
Let's verify this solution:
1. Substitute into the first inequality [tex]\( x - y \leq 3 \)[/tex]:
- [tex]\( -6 - 1 \leq 3 \)[/tex] ⟹ [tex]\( -7 \leq 3 \)[/tex] ⟹ True
2. Substitute into the second inequality [tex]\( 2x + 3y \leq -9 \)[/tex]:
- [tex]\( 2(-6) + 3(1) \leq -9 \)[/tex] ⟹ [tex]\( -12 + 3 \leq -9 \)[/tex] ⟹ -9 \leq -9 ⟹ True
Thus, [tex]\( (x, y) = (-6, 1) \)[/tex] satisfies both inequalities, confirming it as a valid solution.
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