Which system has no solution?
$\begin{array}{c||c||c||c}
y=-3 x+8, & y=9 x+6.25 & y=4.5 x-5 & y=3 x+9 \\6 x+2 y=-4.5 & -18 x+2 y=12.5 & -3 x+2 y=6 & x+8 y=12.3\end{array}$
Answer (2)
Rewrite each system of equations in the form a x + b y = c .
Check the ratios of the coefficients of x and y to determine if the lines are parallel.
System 1 has 6 3 = 2 1 = − 4.5 8 , indicating no solution.
The system with no solution is System 1: y = − 3 x + 8 , 6 x + 2 y = − 4.5 .
Explanation
Understanding the Problem We are given four systems of linear equations and we need to determine which system has no solution. A system of linear equations has no solution if the lines are parallel but have different y-intercepts. This means that the ratio of the coefficients of x and y are equal, but the ratio of the constants is different.
Analyzing System 1 Let's analyze each system:
System 1: y = − 3 x + 8 can be rewritten as 3 x + y = 8 .
The second equation is 6 x + 2 y = − 4.5 .
We check the ratios: 6 3 = 2 1 and 2 1 . However, − 4.5 8 = − 9 16 ≈ − 1.78 . Since 6 3 = 2 1 = − 4.5 8 , this system has no solution.
Analyzing System 2 System 2: y = 9 x + 6.25 can be rewritten as − 9 x + y = 6.25 .
The second equation is − 18 x + 2 y = 12.5 .
We check the ratios: − 18 − 9 = 2 1 and 2 1 . Also, 12.5 6.25 = 2 1 . Since − 18 − 9 = 2 1 = 12.5 6.25 , this system has infinitely many solutions.
Analyzing System 3 System 3: y = 4.5 x − 5 can be rewritten as − 4.5 x + y = − 5 .
The second equation is − 3 x + 2 y = 6 .
We check the ratios: − 3 − 4.5 = 2 3 = 1.5 and 2 1 . Since − 3 − 4.5 = 2 1 , this system has a unique solution.
Analyzing System 4 System 4: y = 3 x + 9 can be rewritten as − 3 x + y = 9 .
The second equation is x + 8 y = 12.3 .
We check the ratios: 1 − 3 = − 3 and 8 1 . Since 1 − 3 = 8 1 , this system has a unique solution.
Conclusion Therefore, the system with no solution is System 1.
Examples
Understanding systems of equations is crucial in various real-world applications. For instance, in economics, it helps determine market equilibrium by analyzing supply and demand curves. In engineering, it's used to solve network problems, such as electrical circuits or traffic flow. Moreover, systems of equations are fundamental in computer graphics for transformations and projections, ensuring objects are rendered correctly on the screen. By mastering these concepts, students gain valuable tools for analyzing and solving complex problems across diverse fields.
The system with no solution is System 1: y = − 3 x + 8 and 6 x + 2 y = − 4.5 because the lines are parallel with equal coefficients but different constants. Thus, they do not intersect at any point.
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