solve graphically. Estimate to the nearest tenth if the roots are not integers.
x^2 y^2
--- + ---- =1
9 4
3x-2y=6
To solve the system of equations graphically, we need to determine the points where the two equations intersect.
Let's start by graphing the first equation, which is an ellipse. To do this, we need to rearrange the equation in the form:
(x^2/9) + (y^2/4) = 1
This equation represents an ellipse centered at the origin with a major axis of length 2a = 6 (since a = 3) along the x-axis and a minor axis of length 2b = 4 (since b = 2) along the y-axis.
To graph the ellipse, draw the coordinate axes on a piece of graph paper. Find the points on the x-axis where x = -3 and x = 3, and mark them. Similarly, find the points on the y-axis where y = -2 and y = 2, and mark them. These points will help you draw the ellipse accurately.
Now, draw the ellipse by connecting the points (-3, 0), (3, 0), (0, -2), and (0, 2) smoothly.
Next, let's graph the second equation, which is a straight line. We can rearrange this equation into slope-intercept form (y = mx + b) to determine its slope and y-intercept.
3x - 2y = 6
-2y = -3x + 6
y = (3/2)x - 3
The equation y = (3/2)x - 3 represents a line with a slope of 3/2 (or 1.5) and a y-intercept of -3.
Now, draw the line on the same graph. Plot the y-intercept at (0, -3), and then use the slope to find more points. For instance, if x = -2, then y = (3/2)(-2) - 3 = -6 - 3 = -9. Plot the point (-2, -9). Similarly, if x = 2, then y = (3/2)(2) - 3 = 3 - 3 = 0. Plot the point (2, 0). Connect all the points to draw the line.
The points where the ellipse and the line intersect are the solutions to the system of equations. These points represent the x and y values that satisfy both equations simultaneously.
Using the graph, estimate the coordinates of the points of intersection to the nearest tenth. Read the x and y values from the graph at the points of intersection and round them to the nearest tenth.
For example, if the intersection point looks close to (2.3, -1.7), then estimate the coordinates as (2.3, -1.7) to the nearest tenth.
Repeat this process for each intersection point to estimate their coordinates. These coordinates represent the approximate solutions to the system of equations.