Select all the correct systems of equations. Which systems of equations have no solution?
[tex]
\begin{aligned}
x+4 y & =23 \
-3 x & =12 y+1
\end{aligned}
[/tex]
[tex]
\begin{aligned}
2 x+y & =17 \
-4 x & =2 y-34
\end{aligned}
[/tex]
[tex]
\begin{aligned}
2 x+4 y & =22 \
-x & =2 y-11
\end{aligned}
[/tex]
[tex]
\begin{aligned}
3 y & =10-x \
2 x+6 y & =7
\end{aligned}
[/tex]
[tex]
\begin{aligned}
2 x+y & =15 \
x & =15-2 y
\end{aligned}
[/tex]
[tex]
\begin{aligned}
y & =13-2 x \
4 x-y & =-1
\end{aligned}
[/tex]
Answer (2)
System 1 has no solution because the equations represent parallel lines: x + 4 y = 23 and − 3 x − 12 y = 1 are parallel.
System 2 has infinite solutions because the equations are identical: 2 x + y = 17 and − 4 x − 2 y = − 34 are the same line.
System 3 has infinite solutions because the equations are identical: 2 x + 4 y = 22 and − x − 2 y = − 11 are the same line.
System 4 has no solution because the equations represent parallel lines: x + 3 y = 10 and 2 x + 6 y = 7 are parallel.
System 5 has a unique solution.
System 6 has a unique solution.
The systems with no solution are System 1 and System 4. S ys t e m 1 , S ys t e m 4
Explanation
Understanding the Problem We are given six systems of linear equations and we need to determine which of them have no solution. A system of linear equations has no solution if the equations represent parallel lines that never intersect. This occurs when the coefficients of the variables are proportional, but the constant terms are not.
Analyzing System 1 Let's analyze each system:
System 1:
x + 4 y − 3 x = 23 = 12 y + 1
Rewrite the second equation as − 3 x − 12 y = 1 . Multiply the first equation by − 3 to get − 3 x − 12 y = − 69 . Comparing this with the second equation, we have − 3 x − 12 y = − 69 and − 3 x − 12 y = 1 . Since the left sides are the same but the right sides are different ( − 69 = 1 ), this system has no solution.
Analyzing System 2 System 2:
2 x + y − 4 x = 17 = 2 y − 34
Rewrite the second equation as − 4 x − 2 y = − 34 . Multiply the first equation by − 2 to get − 4 x − 2 y = − 34 . Comparing this with the second equation, we see that they are identical. This means the system has infinitely many solutions.
Analyzing System 3 System 3:
2 x + 4 y − x = 22 = 2 y − 11
Rewrite the second equation as − x − 2 y = − 11 . Multiply the second equation by − 2 to get 2 x + 4 y = 22 . Comparing this with the first equation, we see that they are identical. This means the system has infinitely many solutions.
Analyzing System 4 System 4:
3 y 2 x + 6 y = 10 − x = 7
Rewrite the first equation as x + 3 y = 10 . Multiply the first equation by 2 to get 2 x + 6 y = 20 . Comparing this with the second equation, we have 2 x + 6 y = 20 and 2 x + 6 y = 7 . Since the left sides are the same but the right sides are different ( 20 = 7 ), this system has no solution.
Analyzing System 5 System 5:
2 x + y x = 15 = 15 − 2 y
Rewrite the second equation as x + 2 y = 15 . We can solve this system using substitution or elimination. Since the equations are not multiples of each other, this system has a unique solution.
Analyzing System 6 System 6:
y 4 x − y = 13 − 2 x = − 1
Rewrite the first equation as 2 x + y = 13 . We can solve this system using substitution or elimination. Since the equations are not multiples of each other, this system has a unique solution.
Final Answer Therefore, the systems with no solution are System 1 and System 4.
Examples
Understanding systems of equations is crucial in various real-world applications. For instance, consider a scenario where you're trying to optimize a budget. You have two different investment options, each with its own rate of return and initial cost. By setting up a system of equations, you can determine the optimal amount to invest in each option to maximize your overall return while staying within your budget. Similarly, systems of equations are used in engineering to design structures, in economics to model supply and demand, and in computer science for network optimization.
The systems with no solution are System 1 and System 4 because they represent parallel lines that never intersect. System 1's equations become inconsistent upon comparison, and System 4 also leads to different results on the same side of an equation, indicating no point of intersection. Thus, the answer is System 1, System 4 .
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