The equations in the system below are parallel.
[tex]
\begin{array}{l}
y=3 x+5 \\
y=3 x+8
\end{array}
[/tex]
How many solutions does the system have?
A. no solution
B. one unique solution
C. two solutions
D. an infinite number of solutions
Answer (2)
The system consists of two linear equations: y = 3 x + 5 and y = 3 x + 8 .
The lines are parallel because they have the same slope (3) but different y-intercepts (5 and 8).
Parallel lines never intersect, meaning there is no point (x, y) that satisfies both equations.
Therefore, the system has no solution .
Explanation
Analyze the problem We are given a system of two linear equations:
y = 3 x + 5
y = 3 x + 8
We need to determine how many solutions this system has, given that the lines are parallel.
Confirm that the lines are parallel Parallel lines have the same slope but different y-intercepts. In this case, both lines have a slope of 3, but the first line has a y-intercept of 5, and the second line has a y-intercept of 8. Since the slopes are the same and the y-intercepts are different, the lines are indeed parallel.
Determine the number of solutions Parallel lines never intersect. A solution to a system of equations is a point (x, y) that satisfies both equations simultaneously. Graphically, this is the point where the two lines intersect. Since parallel lines never intersect, there is no point that satisfies both equations. Therefore, the system has no solution.
State the final answer The system of equations has no solution because the lines are parallel and do not intersect.
Examples
Imagine you're drawing two lines on a piece of paper. If the lines are parallel, they will never meet, no matter how far you extend them. Similarly, in a system of equations, if the equations represent parallel lines, there's no common point that satisfies both equations simultaneously. This concept is useful in various fields, such as urban planning (ensuring roads don't intersect without proper connections) or designing electrical circuits (avoiding short circuits by keeping certain paths separate).
The system of equations has no solutions because the two equations represent parallel lines that do not intersect. Therefore, the correct answer is A. no solution.
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