Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To determine which graph represents the solution to the system of inequalities:
[tex]\[ \begin{array}{l} x + y < 4 \\ 2x - 3y \geq 12 \end{array} \][/tex]
we need to understand and plot each inequality separately and then find the region where both conditions are satisfied.
### Step 1: Plotting [tex]\( x + y < 4 \)[/tex]
1. Rewrite the inequality as an equation to plot the boundary line:
[tex]\[ x + y = 4 \][/tex]
This is a straight line. To find points on this line, we can choose two points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 4 \)[/tex]. Hence, the point is (0, 4).
- When [tex]\( y = 0 \)[/tex], [tex]\( x = 4 \)[/tex]. Hence, the point is (4, 0).
2. Draw the line passing through these two points (0,4) and (4,0).
3. Since the inequality is [tex]\( x + y < 4 \)[/tex], shade the region below the line because this inequality represents all points where the sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is less than 4.
### Step 2: Plotting [tex]\( 2x - 3y \geq 12 \)[/tex]
1. Rewrite the inequality as an equation to plot the boundary line:
[tex]\[ 2x - 3y = 12 \][/tex]
This is a straight line. To find points on this line, we can choose:
- When [tex]\( x = 0 \)[/tex], solving for [tex]\( y \)[/tex]: [tex]\( -3y = 12 \)[/tex] gives [tex]\( y = -4 \)[/tex]. So, the point is (0, -4).
- When [tex]\( y = 0 \)[/tex], solving for [tex]\( x \)[/tex]: [tex]\( 2x = 12 \)[/tex] gives [tex]\( x = 6 \)[/tex]. So, the point is (6, 0).
2. Draw the line passing through these two points (0, -4) and (6, 0).
3. Since the inequality is [tex]\( 2x - 3y \geq 12 \)[/tex], shade the region above the line because this inequality represents all points where [tex]\( 2x - 3y \)[/tex] is greater than or equal to 12.
### Step 3: Finding the solution region
The solution to the system of inequalities is the region where both shaded areas overlap. Graphically:
- The line [tex]\( x + y = 4 \)[/tex] bounds the region below it for [tex]\( x + y < 4 \)[/tex].
- The line [tex]\( 2x - 3y = 12 \)[/tex] bounds the region above it for [tex]\( 2x - 3y \geq 12 \)[/tex].
Find the intersection of the two boundaries:
Set [tex]\( x + y = 4 \)[/tex] and [tex]\( 2x - 3y = 12 \)[/tex]. Solve these equations simultaneously as a system of linear equations:
1. From [tex]\( x + y = 4 \)[/tex], solve for [tex]\( y \)[/tex]:
[tex]\[ y = 4 - x \][/tex]
2. Substitute [tex]\( y = 4 - x \)[/tex] into [tex]\( 2x - 3y = 12 \)[/tex]:
[tex]\[ 2x - 3(4 - x) = 12 \\ 2x - 12 + 3x = 12 \\ 5x - 12 = 12 \\ 5x = 24 \\ x = \frac{24}{5} = 4.8 \][/tex]
3. Substitute [tex]\( x = 4.8 \)[/tex] back into [tex]\( y = 4 - x \)[/tex]:
[tex]\[ y = 4 - 4.8 = -0.8 \][/tex]
The intersection point is [tex]\( (4.8, -0.8) \)[/tex].
### Conclusion
The region where both shading overlap is bounded by the lines with the intersection at [tex]\( (4.8, -0.8) \)[/tex]:
- Below [tex]\( x + y < 4 \)[/tex].
- Above [tex]\( 2x - 3y \geq 12 \)[/tex].
Based on the described constraints and without visual aids, the correct graph should show a shaded region below the line [tex]\( x + y = 4 \)[/tex] and above the line [tex]\( 2x - 3y = 12 \)[/tex], with intersection at the point [tex]\( (4.8, -0.8) \)[/tex].
Thus, you should choose the graph that represents this overlapping region as the solution. Without actual graph options provided as stated "A. [tex]$\qquad$[/tex] B. [tex]\(y \mp 10\)[/tex]," which seems incomplete, we can't specify which graphic is the accurate match. Thus, please verify against the available graphs accordingly.
[tex]\[ \begin{array}{l} x + y < 4 \\ 2x - 3y \geq 12 \end{array} \][/tex]
we need to understand and plot each inequality separately and then find the region where both conditions are satisfied.
### Step 1: Plotting [tex]\( x + y < 4 \)[/tex]
1. Rewrite the inequality as an equation to plot the boundary line:
[tex]\[ x + y = 4 \][/tex]
This is a straight line. To find points on this line, we can choose two points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 4 \)[/tex]. Hence, the point is (0, 4).
- When [tex]\( y = 0 \)[/tex], [tex]\( x = 4 \)[/tex]. Hence, the point is (4, 0).
2. Draw the line passing through these two points (0,4) and (4,0).
3. Since the inequality is [tex]\( x + y < 4 \)[/tex], shade the region below the line because this inequality represents all points where the sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is less than 4.
### Step 2: Plotting [tex]\( 2x - 3y \geq 12 \)[/tex]
1. Rewrite the inequality as an equation to plot the boundary line:
[tex]\[ 2x - 3y = 12 \][/tex]
This is a straight line. To find points on this line, we can choose:
- When [tex]\( x = 0 \)[/tex], solving for [tex]\( y \)[/tex]: [tex]\( -3y = 12 \)[/tex] gives [tex]\( y = -4 \)[/tex]. So, the point is (0, -4).
- When [tex]\( y = 0 \)[/tex], solving for [tex]\( x \)[/tex]: [tex]\( 2x = 12 \)[/tex] gives [tex]\( x = 6 \)[/tex]. So, the point is (6, 0).
2. Draw the line passing through these two points (0, -4) and (6, 0).
3. Since the inequality is [tex]\( 2x - 3y \geq 12 \)[/tex], shade the region above the line because this inequality represents all points where [tex]\( 2x - 3y \)[/tex] is greater than or equal to 12.
### Step 3: Finding the solution region
The solution to the system of inequalities is the region where both shaded areas overlap. Graphically:
- The line [tex]\( x + y = 4 \)[/tex] bounds the region below it for [tex]\( x + y < 4 \)[/tex].
- The line [tex]\( 2x - 3y = 12 \)[/tex] bounds the region above it for [tex]\( 2x - 3y \geq 12 \)[/tex].
Find the intersection of the two boundaries:
Set [tex]\( x + y = 4 \)[/tex] and [tex]\( 2x - 3y = 12 \)[/tex]. Solve these equations simultaneously as a system of linear equations:
1. From [tex]\( x + y = 4 \)[/tex], solve for [tex]\( y \)[/tex]:
[tex]\[ y = 4 - x \][/tex]
2. Substitute [tex]\( y = 4 - x \)[/tex] into [tex]\( 2x - 3y = 12 \)[/tex]:
[tex]\[ 2x - 3(4 - x) = 12 \\ 2x - 12 + 3x = 12 \\ 5x - 12 = 12 \\ 5x = 24 \\ x = \frac{24}{5} = 4.8 \][/tex]
3. Substitute [tex]\( x = 4.8 \)[/tex] back into [tex]\( y = 4 - x \)[/tex]:
[tex]\[ y = 4 - 4.8 = -0.8 \][/tex]
The intersection point is [tex]\( (4.8, -0.8) \)[/tex].
### Conclusion
The region where both shading overlap is bounded by the lines with the intersection at [tex]\( (4.8, -0.8) \)[/tex]:
- Below [tex]\( x + y < 4 \)[/tex].
- Above [tex]\( 2x - 3y \geq 12 \)[/tex].
Based on the described constraints and without visual aids, the correct graph should show a shaded region below the line [tex]\( x + y = 4 \)[/tex] and above the line [tex]\( 2x - 3y = 12 \)[/tex], with intersection at the point [tex]\( (4.8, -0.8) \)[/tex].
Thus, you should choose the graph that represents this overlapping region as the solution. Without actual graph options provided as stated "A. [tex]$\qquad$[/tex] B. [tex]\(y \mp 10\)[/tex]," which seems incomplete, we can't specify which graphic is the accurate match. Thus, please verify against the available graphs accordingly.
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
