To determine the number of solutions for the given system of equations:

Given equations:

1. $-2x + y = 1$
2. $y = -2x + 4$

Step 1: Recognize the equations

Notice that both equations are in the form of a linear equation. We can examine the slopes and y-intercepts to determine if they are parallel, coincident, or intersecting.

Step 2: Compare slopes

Rewrite the first equation to clearly identify its slope and y-intercept.

1. Equation 1: $-2x + y = 1$ Rearrange to solve for $y$: $y = 2x + 1$

   Now, the first equation becomes: $y = 2x + 1$ The slope is $2$, and the y-intercept is $1$.

2. Equation 2: (already in slope-intercept form) $y = -2x + 4$ The slope here is $-2$, and the y-intercept is $4$.

Step 3: Determine if they are parallel, coincident, or intersecting

• If two lines have the same slope but different y-intercepts, they are parallel and have no solutions (no intersection points).
• If two lines have the same slope and the same y-intercept, they are coincident and have infinitely many solutions (all points on one line are also on the other).
• If two lines have different slopes, they intersect at exactly one point (a unique solution).

Step 4: Analyze the slopes and y-intercepts

• The slope of the first equation ($y = 2x + 1$) is $2$.
• The slope of the second equation ($y = -2x + 4$) is $-2$.

Since the slopes are different ($2$ and $-2$), the lines are not parallel. Therefore, these lines will intersect at exactly one point, meaning there is one unique solution to this system of equations.

Conclusion:

The system of equations has exactly one solution.

Summary:

1. Compare slopes and y-intercepts.
2. Different slopes indicate one solution (an intersection point).
3. Same slope with different y-intercepts indicates no solution.
4. Same slope and y-intercept indicate infinitely many solutions.

Would you like further details on solving for the exact intersection point or have any questions?

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Here are five related questions that could expand on this topic:

1. How can we find the exact intersection point for these two lines?
2. What is the graphical interpretation of a system with no solutions?
3. How can we use substitution or elimination to solve a system of equations?
4. How does changing the slope affect the number of solutions in a system?
5. Can systems of nonlinear equations have similar classifications for solutions?

Tip: When comparing two linear equations, always start by analyzing their slopes to quickly assess the number of possible solutions.