1. Solving Systems of Linear Equations and Inequalities by Graphing
Find the x and y intercepts for both equations, graph and connect each pair.
1) -2x + y = -3
2) x + 3y = 5
Equation 1: x = 1.5 and y = -3
Equation 2: x = 5 and y = 1.66
The solution is the point where the two lines intersect
The solution appears to be (2, 1); the exact solution can be obtained by substitution or elimination. At best the solution can only be estimated by graphing.
Comment: Two straight lines could be parallel (no solution) or even the same line (infinite solutions) as well as intersect at the same point (one solution).
Inequalities require that one side of each line be shaded. Select a point. (0, 0) is best (unless it is on the line being tested),
1) -2x + y  -3
2) x + 3y  5
(2, 1)
plug 0 in for the x and y in equation 1 giving 0  -3, since this is not true shade the side of equation 1 that does not contain the point (0, 0).
Do the same for equation 2 getting 0  5, also not true so shade the other side
The solution is the common shaded region,
plus all corner points

2. Systems of inequalities can have more than two inequalities
Solutions to these systems require the combined shaded region and all corner points
Find the x and y intercepts for all equations, graph and connect each pair; remember equation 3 gives a vertical line.
1) x - y  ) 2x + 4y  ) x  -1
(-1, 1.75)
Equation 1: x = 3 and y = -3; plot and connect
(2.66, -.33)
(0, 0) makes it true so shade top side.
(-1, -4)
Equation 2: x = 2 and y = 1; plot and connect
(0, 0) makes it true so shade bottom side.
Multiplying equation 1 by 4 and adding it to equation 2 gives 6x = 16, so x = 8/3 = Substituting that in for x in equation 1 and solving for y gives y = -1/3 = Therefore the corner point where equations 1 & 2 meet is (2.66, -.33)
Equation 3: x = -1 and no y-intercept; plot and draw a vertical line
(0, 0) makes it true too, since 0 is bigger than –1, so shade to the right
Since (0, 0) made all three equations true the region common to all contains (0, 0)
and it has three corners: (2.66, -.33) the intersection of equations 1& 2, and since equation 3 has x = -1, it intersects equation 1 at the point (-1, -4) and equation 2 at the point (-1, 1.75)