Answer:
See the attachments. The first four are graphed in the first attachment; problem 5 is graphed in the second attachment.
The system of equations has 1 solution.
Step-by-step explanation:
1) The equation is in slope-intercept form. The slope is the coefficient of x, which is -2. The y-intercept is the constant term, 3, meaning the point (0, 3) on the y-axis is a point on the graph. From there, or anywhere else on the line, the graph goes down 2 units for each unit to the right. That is, (1, 1) will also be on the graph of this line. Draw a line through these two points to make your graph.
2) The equation is in point-slope form, y - k = m(x - h). In this form, the number outside parentheses (m) is the slope, and the numbers (h, k) are a point on the line. This equation has (h, k) = (-1, 5), so that point is a good place to start your graph. The slope of -3 tells you the line goes down 3 units for each 1 unit to the right, so (0, 2) will be another point on the line. Draw a line through these two points to make your graph.
3) This inequality is written in standard form. For graphing purposes, you can divide by the constant on the right to put it in "intercept form":
... x/x-intercept + y/y-intercept > 1
Doing this, you get
... x/2 + y/6 > 1
This inequality tells you several things about the graph. The line that forms the boundary of the solution area will pass through x-intercept (2, 0) and y-intercept (0, 6). The "greater than" symbol (>) tells you this line is <em>not</em> part of the solution, so will be a dashed line. It also tells you that values of x and y that are above and to the right of the line are part of the solution, so that half-plane is the one you shade to show the solution set. In the attachment, the shading is green.
4) The equation y=1 is the equation of the horizontal line through any point with a y-coordinate of -1. The "less than or equal to" symbol (≤) tells you the line <em>is</em> part of the solution, so will be a solid line. It also tells you that values of y that are below that line are part of the solution, so that half-plane is the one you shade to show the solution set. In the attachment, the shading is purple.
5) Both equations are in slope-intercept form, so can be graphed as described above. Their slopes are different, so there will be exactly one solution.
You were not asked to solve the equations. If you did, you'd find the solution to be (x, y) = (-8/9, -2 5/9).
