Question

if a system of equations results in a statement like 0 = 5, what does it indicate?
a. the system has infinitely many solutions.
b. the system has no solution.
c. the equations are undefined.
d. the system has one solution.
what does it mean if two lines in a system of equations are parallel?
a. they intersect at multiple points.
b. they have no points of intersection.
c. they intersect at one point.
d. they are the same line.
what is the result of eliminating y from the system x + y = 8 and x - y = 2 ?
a. y = 2
b. y = 5
c. x = 5
d. x = 3

Explanation:

Step1: Analyze the first question

A statement like \(0 = 5\) is a contradiction. In a system of equations, if we get a contradiction after solving, it means there is no set of values for the variables that can satisfy all the equations simultaneously. So, the system has no solution.

Step2: Analyze the second question

Parallel lines never meet. In the context of a system of linear equations (where each equation represents a line), if two lines are parallel, they have no common point. That is, there is no solution (no point \((x,y)\) that lies on both lines), which means they have no points of intersection.

Step3: Analyze the third question

We have the system \(

$$\begin{cases}x + y=8\\x - y=2\end{cases}$$

\). If we add the two equations together (\((x + y)+(x - y)=8 + 2\)), the \(y\) - terms cancel out (\(y+( - y)=0\)). So, \(x + y+x - y=8 + 2\) simplifies to \(2x=10\), and then \(x = 5\).

Answer:

1. B. The system has no solution.
2. B. They have no points of intersection.
3. C. \(x = 5\)