Solve each linear system by graphing.
a) x y 3 d) 2x y 10
x y 7 y x 2
b) x y 8 e) y 3x – 5
4x 2y 8 y 2x 5
c) y 2x – 4 f) 6x 5y 12 0
3x y 6 2x 5y 2 0
a) The system of equations is:
1) x + y = 3
2) x + y = 7
To graph these equations, we can rewrite them in slope-intercept form:
1) y = -x + 3
2) y = -x + 7
From the equations, we can see that the two lines have the same slope (-1) but different y-intercepts (3 and 7). Therefore, the lines are parallel and will never intersect.
b) The system of equations is:
1) x + y = 8
2) 4x + 2y = 8
To graph these equations, we can rewrite them in slope-intercept form:
1) y = -x + 8
2) y = -2x + 4
From the equations, we can see that the two lines have different slopes (-1 and -2) and different y-intercepts (8 and 4). Therefore, the lines will intersect at a single point.
c) The system of equations is:
1) y = 2x - 4
2) 3x + y = 6
To graph these equations, we can rewrite them in slope-intercept form:
1) y = 2x - 4
2) y = -3x + 6
From the equations, we can see that the two lines have different slopes (2 and -3) and different y-intercepts (-4 and 6). Therefore, the lines will intersect at a single point.
d) The system of equations is:
1) 2x + y = 10
2) y = x + 2
To graph these equations, we can rewrite them in slope-intercept form:
1) y = -2x + 10
2) y = x + 2
From the equations, we can see that the two lines have different slopes (-2 and 1) and different y-intercepts (10 and 2). Therefore, the lines will intersect at a single point.
e) The system of equations is:
1) y = 3x - 5
2) y = 2x + 5
To graph these equations, we can rewrite them in slope-intercept form:
1) y = 3x - 5
2) y = 2x + 5
From the equations, we can see that the two lines have different slopes (3 and 2) and different y-intercepts (-5 and 5). Therefore, the lines will intersect at a single point.
f) The system of equations is:
1) 6x + 5y = 12
2) 2x + 5y = 2
To graph these equations, we can rewrite them in slope-intercept form:
1) y = -6/5x + 12/5
2) y = -2/5x + 2/5
From the equations, we can see that the two lines have different slopes (-6/5 and -2/5) and different y-intercepts (12/5 and 2/5). Therefore, the lines will intersect at a single point.