Question 2: Infinite Solutions
Consider the two equations you circled, which form a system with infinite solutions.
Graph them together on the following coordinate grid.
Plug a few points into the original equations and check to show that the system of
equations has more than one solution. Explain your work.
Equation 1: y = 2x + 3
Equation 2: -2x + 2y = 6
To graph these equations, we first need to rewrite Equation 2 in slope-intercept form:
2y = 2x + 6
y = x + 3
Now, we can graph both equations on the same coordinate grid:
Equation 1 (y = 2x + 3):
- Start by plotting the y-intercept at (0, 3)
- Use the slope of 2 to find another point, such as (1, 5) and (-1, 1)
- Draw a straight line connecting these points
Equation 2 (-2x + 2y = 6 or y = x + 3):
- Plot the y-intercept at (0, 3)
- Use the slope of 1 to find another point, such as (1, 4) and (-1, 2)
- Draw a straight line connecting these points
After graphing both equations, you will see that the lines are overlapping. This indicates that the system of equations has infinite solutions. To confirm this, you can plug in a few points into both equations. For example:
- Plug in x = 1 into Equation 1: y = 2(1) + 3 = 5
- Plug in x = 1 into Equation 2: -2(1) + 2(5) = 6
As the points (1, 5) and (1, 6) satisfy both equations, it confirms that the system has infinite solutions.