Solve the system by graphing. (If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.)
x − y = 2
2x + 3y = 19
To solve this system of equations by graphing, we can plot the lines representing each equation on a coordinate plane and find their intersection point. The intersection point will be the solution to the system.
First, let's rearrange the equations in slope-intercept form (y = mx + b) to make it easier to graph them:
Equation 1: x - y = 2
Rearranging, we get y = x - 2
Equation 2: 2x + 3y = 19
Rearranging, we get y = (19 - 2x) / 3
Now, we can plot these two lines on a graph. Choose a suitable range for the x and y axes, such as -5 to 5, and mark the points for each line using the respective equations we obtained:
For Equation 1:
- When x = 0, y = -2, so plot the point (0, -2).
- When x = 2, y = 0, so plot the point (2, 0).
For Equation 2:
- When x = 0, y = 19/3 ≈ 6.33, so plot the point (0, 6.33).
- When x = 8, y = 1/3, so plot the point (8, 1/3 ≈ 0.33).
Now, connect the points for each line using a straight line. The lines should intersect at a specific point.
Once you have graphed the lines, determine the coordinates of the intersection point (if they intersect). The coordinates of the intersection point represent the solution of the system.
If the lines never intersect and are parallel, the system is inconsistent and has no solution.
If the lines are coincident or overlapping, meaning they intersect at infinitely many points, the system is dependent and has infinitely many solutions.
Therefore, analyze the graph and determine whether the lines intersect at a single point, are parallel, or coincide.