Question

1. how many solutions does the system have and what does it look like graphically? $-2x + y = 1$ and $2x + y = 2$

a. no solution
b. one solution
c. infinite solutions
d. parallel lines
e. intersecting lines
f. single line

Explanation:

Step1: Analyze the slopes of the lines

First, we can rewrite the two equations in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

For the equation \(-2x + y=1\), we can solve for \(y\):
\(y = 2x + 1\). The slope \(m_1\) of this line is \(2\).

For the equation \(2x + y = 2\), we solve for \(y\):
\(y=-2x + 2\). The slope \(m_2\) of this line is \(- 2\).

Since \(m_1
eq m_2\) (the slopes are not equal), the two lines are not parallel. Also, since the slopes are not equal, the two lines are not the same line (so there are not infinite solutions).

Step2: Determine the number of solutions

A system of linear equations \(a_1x + b_1y=c_1\) and \(a_2x + b_2y = c_2\) has one solution when \(\frac{a_1}{a_2}
eq\frac{b_1}{b_2}\) (in the case of non - parallel lines). In our case, for the equations \(-2x + y = 1\) (or \(2x - y=- 1\)) and \(2x + y = 2\), \(\frac{-2}{2}
eq\frac{1}{1}\) (comparing the coefficients of \(x\) and \(y\)). So the two lines will intersect at exactly one point, which means the system has one solution. And graphically, two lines with different slopes are intersecting lines.

Answer:

B. One Solution, E. Intersecting Lines