Without graphing, classify each system of equations as independent, dependent, or inconsistent. Solve independent systems by graphing.
{1 + y = x
{ x + y = 1
The answer is Independent (1,0); could you show how to get this answer step by step?
To solve this system of equations by graphing, you need to graph the two lines represented by the equations and find their point of intersection. If the lines intersect at a single point, the system is independent; if they coincide and cover the same points, the system is dependent; and if they are parallel and never intersect, the system is inconsistent.
First, let's solve the first equation for x to get it in slope-intercept form (y = mx + b):
1 + y = x ----> x = y + 1
Next, let's solve the second equation for y:
x + y = 1 ----> y = 1 - x
Now you can graph both lines:
For the first equation, use the slope-intercept form: x = y + 1
To graph a line using slope-intercept form, start by plotting the y-intercept, which is the point (0, 1). From there, use the slope, which is 1, to determine additional points on the line. Since the slope is 1 (which can be written as 1/1), you can move one unit up and one unit to the right to find the next point. Plot that point. Repeat this process as needed.
For the second equation, use the slope-intercept form: y = 1 - x
Similarly, plot the y-intercept at (0, 1) and use the slope of -1 to find additional points. The slope of -1 means you can move one unit down and one unit to the right to find the next point. Plot that point. Repeat as needed.
The two lines intersect at the point (1, 0). Since they intersect at a single point, the system is independent.