Solve the system graphically. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this.
2x – y = 4
5x – y = 13
I do not understand how to do this
https://www.wolframalpha.com/input/?i=plot+y%3D2x-4+and+y%3D5x-13
http://www.wolframalpha.com/input/?i=plot+y%3D2x-4+and+y%3D5x-13
x = 3 , y = 2 is the intersection
To solve the system of equations graphically, you need to plot the equations on a graph and find the point of intersection (if any). Let's go step by step:
1. Rearrange both equations in the slope-intercept form, y = mx + b, where "m" is the slope and "b" is the y-intercept.
Equation 1: 2x - y = 4
Rearranging gives: y = 2x - 4
Equation 2: 5x - y = 13
Rearranging gives: y = 5x - 13
2. Choose a suitable range for each axis on the graph. It's a good idea to choose values that will help you easily plot and interpret the points. For simplicity, let's use a range from -5 to 5 for both x and y.
3. Plot the lines on the graph. To do this:
- Pick a couple of values for x, plug them into each equation one at a time, and find the corresponding y-values.
- Plot the points on the graph.
- Draw a straight line passing through the points.
For Equation 1 (y = 2x - 4):
Let's choose x = -2:
When x = -2, y = 2(-2) - 4 = -4 - 4 = -8
Plot the point (-2, -8)
Let's choose x = 2:
When x = 2, y = 2(2) - 4 = 4 - 4 = 0
Plot the point (2, 0)
Draw a line passing through (-2, -8) and (2, 0). This is the graph of Equation 1.
For Equation 2 (y = 5x - 13):
Let's choose x = -1:
When x = -1, y = 5(-1) - 13 = -5 - 13 = -18
Plot the point (-1, -18)
Let's choose x = 3:
When x = 3, y = 5(3) - 13 = 15 - 13 = 2
Plot the point (3, 2)
Draw a line passing through (-1, -18) and (3, 2). This is the graph of Equation 2.
4. Determine the point of intersection (if any). Look for the point where the two lines intersect on the graph.
From the graph, it appears that the two lines intersect at the point (2, 0). This means that (x, y) = (2, 0) is the solution to the system of equations.
Therefore, the solution to the system of equations is x = 2 and y = 0.
In set-builder notation, this would be written as { (x, y) | x = 2, y = 0 }.