Question

example 2: systems with no solutions:
solve by graphing. y = -2x + 1
y = -2x - 1
what do you notice?
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why cant there be a solution?
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without graphing, how can you tell if a
system will have a solution or not?
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Explanation:

1. For the first question: Observe the coefficients of x in both linear equations.
2. For the second question: Parallel lines never intersect, so no shared (x,y) pair exists.
3. For the third question: Compare the slopes and y-intercepts of the linear equations in the system.

Answer:

1. What do you notice?

The slopes (coefficients of $x$) of the two lines are identical ($-2$), while the y-intercepts are different.

1. Why can't there be a solution?

The lines are parallel (same slope, different y-intercepts) and never intersect, so there is no common $(x,y)$ pair that satisfies both equations.

1. Without graphing, how can you tell if a system will have a solution or not?

For a linear system in the form $y=mx+b$:

• If the slopes ($m$) are different, the system has exactly one solution.
• If the slopes are equal but y-intercepts ($b$) are different, the system has no solution (parallel lines).
• If the slopes and y-intercepts are both equal, the system has infinitely many solutions (coinciding lines).