Solve the system by graphing. (If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.)
4x + 2y = 12
5x − 4y = 2
To solve the system by graphing, we will graph each equation on the same coordinate plane and identify the point of intersection (if any).
1. Equations:
- 4x + 2y = 12
- 5x - 4y = 2
2. Convert each equation into slope-intercept form (y = mx + b):
- Equation 1: 4x + 2y = 12 --> 2y = -4x + 12 --> y = -2x + 6
- Equation 2: 5x - 4y = 2 --> -4y = -5x + 2 --> y = (5/4)x - (1/2)
3. Graph each equation:
- Equation 1: y = -2x + 6
- Plot the y-intercept (0,6)
- Find another point: substitute x = 1: y = -2(1) + 6 --> y = 4
- Plot the point (1, 4)
- Draw a straight line through the points
- Equation 2: y = (5/4)x - (1/2)
- Plot the y-intercept (0, -1/2)
- Find another point: substitute x = 1: y = (5/4)(1) - (1/2) --> y = (5/4) - (1/2) --> y = (5/4) - (2/4) --> y = 3/4
- Plot the point (1, 3/4)
- Draw a straight line through the points
4. Determine the point of intersection:
- Look for the point at which the two lines intersect
- If they intersect at a single point, there is one unique solution for the system.
5. Analyzing the graph:
- If the lines intersect at a single point, that is the solution to the system of equations.
- If the lines are parallel and do not intersect, the system is inconsistent and has no solution.
- If the lines are the same (coincide), the system is dependent and has infinitely many solutions.
By analyzing the graph of the given equations, you should be able to determine whether the system is consistent, inconsistent, or dependent.