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Consider the system of linear equations.

y = 34
x + 12

y = 43
x

Part A
How many solutions does the system have?
no solution
exactly one solution
exactly two solutions
infinitely many solutions
Part B
How can you tell?
The slopes of the equations are the same so the lines will not intersect.
The slopes of the equations are different so the lines will intersect at one point.
The slopes of the equations are different so the lines will intersect twice.
The slopes of the equations are the same so the lines will both be the same line.

1 answer

To analyze the given system of equations:

  1. The first equation is: \[ y = 34 \] This represents a horizontal line where \(y\) is always 34, regardless of \(x\).

  2. The second equation is: \[ y = 43 - x \] This can be rewritten in slope-intercept form as \(y = -x + 43\), which has a slope of -1.

Part A: Solutions of the System

Since the first equation is a horizontal line at \(y = 34\) and the second equation has a different slope (specifically, it is a line whose \(y\)-value changes depending on \(x\)), these two lines will not intersect. Therefore, the system of equations has:

Answer: no solution

Part B: Explanation of the Result

The first equation, a horizontal line, has a slope of 0, while the second equation (which has a slope of -1) has a different slope. Since the slopes are different, the lines will not intersect. Therefore, you can conclude:

Answer: The slopes of the equations are different so the lines will not intersect.