Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Let's graph the solution of the given system step by step.
The system of inequalities is:
[tex]\[ \begin{array}{lr} x - 3y \geq -3 \\ 2x + y \leq 6 \end{array} \][/tex]
We will graph each inequality one by one and then determine the region of the graph that satisfies both inequalities.
### Step 1: Graph [tex]\( x - 3y \geq -3 \)[/tex]
1. Convert to equality: Rewrite the inequality as an equation to find the boundary line:
[tex]\[ x - 3y = -3 \][/tex]
2. Find intercepts:
- For the x-intercept (when [tex]\( y = 0 \)[/tex]):
[tex]\[ x - 3(0) = -3 \implies x = -3 \][/tex]
So, the x-intercept is [tex]\((-3, 0)\)[/tex].
- For the y-intercept (when [tex]\( x = 0 \)[/tex]):
[tex]\[ 0 - 3y = -3 \implies y = 1 \][/tex]
So, the y-intercept is [tex]\((0, 1)\)[/tex].
3. Draw the boundary line: Plot the points [tex]\((-3, 0)\)[/tex] and [tex]\((0, 1)\)[/tex] and draw a solid line through them, as the inequality includes the boundary line (≥).
4. Shade the region: To determine which side of the line to shade, use a test point (e.g., [tex]\( (0, 0) \)[/tex]):
[tex]\[ 0 - 3(0) \geq -3 \implies 0 \geq -3 \][/tex]
This is true, so we shade the region that includes the point [tex]\((0, 0)\)[/tex].
### Step 2: Graph [tex]\( 2x + y \leq 6 \)[/tex]
1. Convert to equality: Rewrite the inequality as an equation to find the boundary line:
[tex]\[ 2x + y = 6 \][/tex]
2. Find intercepts:
- For the x-intercept (when [tex]\( y = 0 \)[/tex]):
[tex]\[ 2x + 0 = 6 \implies x = 3 \][/tex]
So, the x-intercept is [tex]\((3, 0)\)[/tex].
- For the y-intercept (when [tex]\( x = 0 \)[/tex]):
[tex]\[ 2(0) + y = 6 \implies y = 6 \][/tex]
So, the y-intercept is [tex]\((0, 6)\)[/tex].
3. Draw the boundary line: Plot the points [tex]\((3, 0)\)[/tex] and [tex]\((0, 6)\)[/tex] and draw a solid line through them, as the inequality includes the boundary line (≤).
4. Shade the region: To determine which side of the line to shade, use a test point (e.g., [tex]\( (0, 0) \)[/tex]):
[tex]\[ 2(0) + 0 \leq 6 \implies 0 \leq 6 \][/tex]
This is true, so we shade the region that includes the point [tex]\( (0, 0) \)[/tex].
### Step 3: Determine the solution region
The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This region satisfies both [tex]\( x - 3y \geq -3 \)[/tex] and [tex]\( 2x + y \leq 6 \)[/tex].
To graph the solution region:
1. Draw the boundary lines based on the intercept points and the information obtained from the inequalities' boundary lines.
2. Shade accordingly: The final region of interest that satisfies both inequalities is the overlapping shaded region from both inequalities.
Here is a sketch of the graph:
- Draw the boundary line for [tex]\( x - 3y = -3 \)[/tex] passing through [tex]\((-3, 0)\)[/tex] and [tex]\((0, 1)\)[/tex], and shade above the line.
- Draw the boundary line for [tex]\( 2x + y = 6 \)[/tex] passing through [tex]\((3, 0)\)[/tex] and [tex]\((0, 6)\)[/tex], and shade below the line.
The intersection of these shaded regions represents the solution to the system.
The system of inequalities is:
[tex]\[ \begin{array}{lr} x - 3y \geq -3 \\ 2x + y \leq 6 \end{array} \][/tex]
We will graph each inequality one by one and then determine the region of the graph that satisfies both inequalities.
### Step 1: Graph [tex]\( x - 3y \geq -3 \)[/tex]
1. Convert to equality: Rewrite the inequality as an equation to find the boundary line:
[tex]\[ x - 3y = -3 \][/tex]
2. Find intercepts:
- For the x-intercept (when [tex]\( y = 0 \)[/tex]):
[tex]\[ x - 3(0) = -3 \implies x = -3 \][/tex]
So, the x-intercept is [tex]\((-3, 0)\)[/tex].
- For the y-intercept (when [tex]\( x = 0 \)[/tex]):
[tex]\[ 0 - 3y = -3 \implies y = 1 \][/tex]
So, the y-intercept is [tex]\((0, 1)\)[/tex].
3. Draw the boundary line: Plot the points [tex]\((-3, 0)\)[/tex] and [tex]\((0, 1)\)[/tex] and draw a solid line through them, as the inequality includes the boundary line (≥).
4. Shade the region: To determine which side of the line to shade, use a test point (e.g., [tex]\( (0, 0) \)[/tex]):
[tex]\[ 0 - 3(0) \geq -3 \implies 0 \geq -3 \][/tex]
This is true, so we shade the region that includes the point [tex]\((0, 0)\)[/tex].
### Step 2: Graph [tex]\( 2x + y \leq 6 \)[/tex]
1. Convert to equality: Rewrite the inequality as an equation to find the boundary line:
[tex]\[ 2x + y = 6 \][/tex]
2. Find intercepts:
- For the x-intercept (when [tex]\( y = 0 \)[/tex]):
[tex]\[ 2x + 0 = 6 \implies x = 3 \][/tex]
So, the x-intercept is [tex]\((3, 0)\)[/tex].
- For the y-intercept (when [tex]\( x = 0 \)[/tex]):
[tex]\[ 2(0) + y = 6 \implies y = 6 \][/tex]
So, the y-intercept is [tex]\((0, 6)\)[/tex].
3. Draw the boundary line: Plot the points [tex]\((3, 0)\)[/tex] and [tex]\((0, 6)\)[/tex] and draw a solid line through them, as the inequality includes the boundary line (≤).
4. Shade the region: To determine which side of the line to shade, use a test point (e.g., [tex]\( (0, 0) \)[/tex]):
[tex]\[ 2(0) + 0 \leq 6 \implies 0 \leq 6 \][/tex]
This is true, so we shade the region that includes the point [tex]\( (0, 0) \)[/tex].
### Step 3: Determine the solution region
The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This region satisfies both [tex]\( x - 3y \geq -3 \)[/tex] and [tex]\( 2x + y \leq 6 \)[/tex].
To graph the solution region:
1. Draw the boundary lines based on the intercept points and the information obtained from the inequalities' boundary lines.
2. Shade accordingly: The final region of interest that satisfies both inequalities is the overlapping shaded region from both inequalities.
Here is a sketch of the graph:
- Draw the boundary line for [tex]\( x - 3y = -3 \)[/tex] passing through [tex]\((-3, 0)\)[/tex] and [tex]\((0, 1)\)[/tex], and shade above the line.
- Draw the boundary line for [tex]\( 2x + y = 6 \)[/tex] passing through [tex]\((3, 0)\)[/tex] and [tex]\((0, 6)\)[/tex], and shade below the line.
The intersection of these shaded regions represents the solution to the system.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
