Question

To solve the system of linear equations 3 x minus 2 y = 4 and 9 x minus 6 y = 12 by using the linear combination method, Henry decided that he should first multiply the first equation by –3 and then add the two equations together to eliminate the x-terms. When he did so, he also eliminated the y-terms and got the equation 0 = 0, so he thought that the system of equations must have an infinite number of solutions. To check his answer, he graphed the equations 3 x minus 2 y = 4 and 9 x minus 6 y = 12 with his graphing calculator, but he could only see one line. Why is this?
because the system of equations actually has only one solution
because the system of equations actually has no solution
because the graphs of the two equations overlap each other
because the graph of one of the equations does not exist

Answer 1

The correct answer is: **because the graphs of the two equations overlap each other**.

To elaborate, the two equations given:

1. \( 3x - 2y = 4 \)
2. \( 9x - 6y = 12 \)

Upon inspection, the second equation can be simplified. If we divide the second equation by 3, we get:

\[
\frac{9x - 6y}{3} = \frac{12}{3} \implies 3x - 2y = 4
\]

This shows that both equations are actually the same. When two equations represent the same line, their graphs will overlap entirely, meaning there are infinitely many solutions (every point on the line is a solution). Therefore, Henry's initial conclusion that there is an infinite number of solutions was correct, but he misinterpreted the elimination process - the resulting equation of \(0 = 0\) is valid and indicates that the two equations are dependent (representing the same line).