How many solutions does this linear system have?
[tex]
\begin{array}{l}
y=2 x-5 \
-8 x-4 y=-20
\end{array}
[/tex]
A. one solution: (-2.5,0)
B. one solution: (2.5,0)
C. no solution
D. infinite number of solutions
Answer (2)
Rewrite the second equation in slope-intercept form to easily compare it with the first equation.
Compare the slopes of the two equations. If the slopes are different, the lines intersect at a single point.
Find the solution by setting the two equations equal to each other and solving for x and y.
The system has one solution: ( 2.5 , 0 ) .
Explanation
Analyze the equations We are given a system of two linear equations:
Equation 1: y = 2 x − 5 Equation 2: − 8 x − 4 y = − 20
Our goal is to determine how many solutions this system has. To do this, we can analyze the equations and see if they represent the same line, parallel lines, or intersecting lines.
Rewrite the second equation Let's rewrite the second equation in slope-intercept form ( y = m x + b ) to easily compare it with the first equation.
− 8 x − 4 y = − 20 − 4 y = 8 x − 20 y = − 2 x + 5
Now we have: Equation 1: y = 2 x − 5 Equation 2: y = − 2 x + 5
Compare slopes Comparing the two equations, we see that the slopes are different. The slope of Equation 1 is 2, and the slope of Equation 2 is -2. Since the slopes are different, the lines intersect at a single point, which means there is exactly one solution to the system.
Find the solution To find the solution, we can set the two equations equal to each other:
2 x − 5 = − 2 x + 5 4 x = 10 x = 4 10 = 2 5 = 2.5
Now, substitute x = 2.5 into either equation to find the value of y . Using Equation 1:
y = 2 ( 2.5 ) − 5 = 5 − 5 = 0
So the solution is ( 2.5 , 0 ) .
Conclusion Since the slopes of the two lines are different, the system has exactly one solution. We found that solution to be x = 2.5 and y = 0 . Therefore, the system has one solution: ( 2.5 , 0 ) .
Examples
Understanding the number of solutions in a linear system is crucial in various real-world applications. For instance, consider a scenario where you're trying to determine the break-even point for two different business plans. Each plan can be represented as a linear equation, with variables representing costs and revenue. If the system has one solution, it indicates a unique break-even point where both plans yield the same profit. If there are infinite solutions, it means the plans are essentially identical in terms of cost and revenue. If there's no solution, the plans will never break even under the given conditions. Analyzing linear systems helps in making informed decisions in business, economics, and engineering.
The linear system has one solution, which is ( 2.5 , 0 ) . The lines represented by the equations intersect at this point because their slopes are different. Thus, the correct choice is B: one solution: (2.5,0).
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