Is the graph correct? What is the classification of the system? If the graph is not correct, give the correct solution.
Hint: Solve each equation for $y$ and check the slopes and $y$-intercepts. If the slopes and $y$-intercepts do not check, then graph each line on paper to find the solution.
$\left\{\begin{array}{l}
9 x+6 y=18 \\
2 y=6-3 x
\end{array}\right.$
A) The graph is NOT correct. The classification is Consistent and Dependent. The correct solution is all real numbers.
B) The graph IS correct. The classification is Consistent and Dependent.
Answer (2)
Solve each equation for y and find that both equations are y = − 2 3 x + 3 .
Since the slopes and y -intercepts are the same, the lines are coincident.
The system is classified as Consistent and Dependent.
The correct solution is all real numbers that satisfy the equation, and the graph IS correct. B
Explanation
Analyze the problem We are given a system of two linear equations:
9 x + 6 y = 18 2 y = 6 − 3 x
We need to determine if the graph is correct and classify the system. If the graph is incorrect, we need to provide the correct solution.
Solve for y Let's solve each equation for y to find the slope and y -intercept.
Equation 1:
9 x + 6 y = 18 6 y = − 9 x + 18 y = 6 − 9 x + 6 18 y = − 2 3 x + 3
Equation 2:
2 y = 6 − 3 x y = − 2 3 x + 3
Compare slopes and intercepts Comparing the slopes and y -intercepts of the two equations, we see that they are the same. Both equations have a slope of − 2 3 and a y -intercept of 3 . This means the lines are coincident (the same line), and the system is consistent and dependent with infinitely many solutions.
Determine the classification and solution Since the lines are coincident, any point on the line is a solution to the system. The solution is all real numbers that satisfy either equation. The classification of the system is Consistent and Dependent. Therefore, the graph IS correct.
Final Answer The graph is correct, and the classification is Consistent and Dependent.
Examples
Consider a situation where you are mixing two solutions with the same ratio of ingredients. If the equations representing the amount of each ingredient are dependent, it means you can scale one solution to obtain the other. This concept is useful in chemistry, cooking, and other fields where maintaining proportions is crucial. For example, if you are making a cleaning solution and the ratio of water to concentrate is the same in two different recipes, the system is dependent, and you can adjust the quantities accordingly.
The graph is correct, and the system is classified as Consistent and Dependent since both equations represent the same line. Thus, the correct solution is all real numbers that satisfy the equations.
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