The system of equations is solved using the linear combination method.
[tex]\begin{array}{l}
\frac{1}{2} x+4 y=8 \rightarrow-2\left(\frac{1}{2} x+4 y=8\right) \rightarrow-x-8 y=-16 \\
3 x+24 y=12 \rightarrow \frac{1}{3}(3 x+24 y=12) \rightarrow \frac{x+8 y=4}{0=-12}
\end{array}[/tex]
What does [tex]$0=-12$[/tex] mean regarding the solution to the system?
A. There are no solutions to the system because the equations represent parallel lines.
B. There are no solutions to the system because the equations represent the same line.
C. There are infinitely many solutions to the system because the equations represent parallel lines.
D. There are infinitely many solutions to the system because the equations represent the same line.
Answer (2)
The linear combination method leads to a contradiction: 0 = − 12 .
Rewrite both equations in slope-intercept form to find the slopes and y-intercepts.
The equations have the same slope but different y-intercepts, indicating parallel lines.
Therefore, there are no solutions to the system because the equations represent parallel lines: There are no solutions to the system because the equations represent parallel lines.
Explanation
Understanding the Result We are given a system of equations that has been manipulated using the linear combination method, resulting in the statement 0 = − 12 . We need to determine what this result means for the solution of the system.
Analyzing the Contradiction The statement 0 = − 12 is a contradiction, which means the original system of equations has no solution. This occurs when the equations represent either parallel lines or inconsistent equations. To determine which is the case, we can rewrite the original equations in slope-intercept form ( y = m x + b ) to compare their slopes and y-intercepts.
Rewriting Equation 1 Let's rewrite the first equation, 2 1 x + 4 y = 8 , in slope-intercept form: 4 y y = − 2 1 x + 8 = − 8 1 x + 2
Rewriting Equation 2 Now, let's rewrite the second equation, 3 x + 24 y = 12 , in slope-intercept form: 24 y y y = − 3 x + 12 = − 24 3 x + 24 12 = − 8 1 x + 2 1
Comparing Slopes and Intercepts Comparing the slope-intercept forms of the two equations, we see that they have the same slope ( − 8 1 ) but different y-intercepts (2 and 2 1 ). This means the equations represent parallel lines.
Conclusion Since the lines are parallel, they never intersect, and therefore, the system of equations has no solution. The contradiction 0 = − 12 indicates that the system is inconsistent, and the equations represent parallel lines.
Examples
Consider you're trying to plan a route using two different map apps. If the apps give you contradictory directions that can never lead to the same destination, it's like having a system of equations with no solution. In this case, the routes are like parallel lines that never meet, meaning you can't satisfy both sets of directions simultaneously. Recognizing such inconsistencies is crucial in real-world problem-solving to avoid wasting time on impossible scenarios.
The contradiction 0 = − 12 from the linear combination method indicates that the system of equations has no solution. This occurs because the two equations represent parallel lines with the same slope but different y-intercepts. Thus, the correct option is (A).
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