Solve each system graphically. Be sure to check your solution.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 = 1
x + 2y = 3
http://www.wolframalpha.com/input/?i=solve+2x+-+y+%3D+1%2C+x+%2B+2y+%3D+3+
To solve the given system of equations graphically, we need to plot the graphs of each equation on the coordinate plane and find their point of intersection. The point of intersection represents the solution to the system.
Let's start with the first equation: 2x - y = 1
To graph this equation, we can rearrange it into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
-y = -2x + 1
y = 2x - 1
Now, let's plot the graph of this equation. First, pick a few values for x and calculate the corresponding y values:
When x = 0, y = 2(0) - 1 = -1
When x = 1, y = 2(1) - 1 = 1
When x = -1, y = 2(-1) - 1 = -3
Plotting these points on the coordinate plane will give us a straight line.
Now, let's move on to the second equation: x + 2y = 3
Again, we can rearrange this equation to slope-intercept form:
2y = -x + 3
y = (-1/2)x + 3/2
Using the same method as before, let's calculate some points for this equation:
When x = 0, y = (-1/2)(0) + 3/2 = 3/2
When x = 1, y = (-1/2)(1) + 3/2 = 1
When x = -1, y = (-1/2)(-1) + 3/2 = 2
Plotting these points will give us another straight line on the coordinate plane.
Now, we need to find the point of intersection between these two lines. This point represents the solution to the system of equations.
After plotting the graphs, we can see that the lines intersect at the point (1, 1). Therefore, the solution to the system is x = 1, y = 1.
To check our solution, we can substitute these values back into the original equations:
For the first equation: 2(1) - 1 = 1, which is true.
For the second equation: 1 + 2(1) = 3, which is also true.
Therefore, our solution is correct.
In conclusion, the system of equations has a unique solution: x = 1, y = 1.