Graph the two lines
2x + 3y = 18
3x -4y > 16.
Give the Domain and Range, Slope, and Y-intercept for each line.
Graph each equation above on the graph below and show all work. Give the Domain and Range, Slope, and Y-intercept for each line. Explain in detail how you got each answer.
2x + 3y = 18 3x -4y > 16
Slope-Intercept Form: Slope-Intercept Form:
Domain: Domain:
Range: Range:
Slope: Slope:
Y-intercept: Y-intercept:
unless the line is vertical or horizontal, it extends left-right and up-down forever.
So, the domain and range are both (-∞,+∞)
the slope of Ax+By=C is -A/B
for the x-intercept, set y=0
for the y-intercept, set x=0
for example the intercepts of 3x+4y=8 are (0,2) and (8/3,0)
for 3x-4y > 16
the solution set is all the shaded area below the line 3x-4y=16.
Thx so much Steve. I think I still need a little help tho.
To graph the lines and find their respective domain, range, slope, and y-intercept, we will first rearrange both equations into slope-intercept form (y = mx + b).
Starting with the equation 2x + 3y = 18:
1. Subtract 2x from both sides: 3y = -2x + 18
2. Divide every term by 3: y = (-2/3)x + 6
Now the equation is in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. From this form, we can identify the slope, y-intercept, and analyze the domain and range.
Slope-Intercept Form: y = (-2/3)x + 6
Slope: The slope of the line is -2/3. It means that for every 3 units moved in the positive x-direction, the line goes down by 2 units in the positive y-direction.
Y-intercept: The y-intercept is 6. It means that the line crosses the y-axis at the point (0, 6).
Domain and Range: Since this is a straight line, the domain and range are both all real numbers, unless specified otherwise.
Now let's analyze the second equation, 3x - 4y > 16:
We will first rearrange it into slope-intercept form, but since there is an inequality, we need to isolate y.
1. Subtract 3x from both sides: -4y > -3x + 16
2. Divide every term by -4, remembering to flip the inequality sign: y < (3/4)x - 4
Now the equation is in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. From this form, we can identify the slope, y-intercept, and analyze the domain and range.
Slope-Intercept Form: y < (3/4)x - 4
Slope: The slope of the line is 3/4. It means that for every 4 units moved in the positive x-direction, the line goes up by 3 units in the positive y-direction.
Y-intercept: The y-intercept is -4. It means that the line crosses the y-axis at the point (0, -4).
Domain and Range: Since this is an inequality, the domain and range are infinite. However, since it is a strict inequality (less than), the points on the line itself are not included in the solution.
Now, let's graph these lines on a coordinate plane and indicate the domain, range, slope, and y-intercept.
Note: Without the actual graph, I will describe the sketch verbally.
For the line 2x + 3y = 18:
- Plot the point (0, 6) as the y-intercept.
- Use the slope as a guide: Go down 2 units and right 3 units to plot another point. Connect the y-intercept and the second point to create the line.
For the line 3x - 4y > 16:
- Plot the point (0, -4) as the y-intercept.
- Use the slope as a guide: Go up 3 units and right 4 units to plot another point. Connect the y-intercept and the second point with a dashed line that does not include the points themselves.
The domain for both lines is all real numbers (-∞, +∞) unless specified otherwise. The range is also all real numbers (-∞, +∞).
For the line 2x + 3y = 18:
- Slope: -2/3
- Y-intercept: (0, 6)
For the line 3x - 4y > 16:
- Slope: 3/4
- Y-intercept: (0, -4)
Remember, this explanation does not include an actual graph, so it is important to sketch it on paper or a graphing software to visualize the lines accurately.