To determine whether the system of equations has solutions, we need to analyze the given equations. The initial expressions you provided seem to represent lines:
- Equation 1: y = 2x
- Equation 2: y = 4x
Both of these equations are in the slope-intercept form (y = mx + b), where m represents the slope and b is the y-intercept. The first equation has a slope of 2, which means for every unit increase in x, y increases by 2 units. The second equation has a slope of 4, meaning y increases by 4 units for each unit increase in x.
To find the solutions where these two equations intersect, we can set them equal to each other:
2x = 4xSolving this yields:
- Subtracting 2x from both sides gives: 0 = 2x
- Therefore, x = 0.
Now, substituting x = 0 back into either equation to find y:
y = 2(0) = 0This indicates that the point (0, 0) is a proposed solution where both lines intersect.
Now, because the slopes of the two lines (2 and 4) are different, the two lines are not parallel, indicating they will intersect at exactly one point. Hence, the system of equations does indeed have a solution, which is:
- Solution: (0, 0)
To summarize, the system of equations has one solution at the point (0, 0). If any additional equations exist in your original query, we would need to consider those as well to analyze the complete system of equations.