Question

Complete the statement with an equation that makes the statement true.(1 point)
The system of equations 3x+4y=1
and y=?
has an infinite number of solutions.

Answer 1

For the system of equations to have an infinite number of solutions, the second equation must be a multiple of the first equation. This means it must represent the same line as the first equation.

Starting with the equation \(3x + 4y = 1\), we can solve it for \(y\):

\[
4y = 1 - 3x \quad \text{(subtracting } 3x \text{ from both sides)}
\]

\[
y = \frac{1 - 3x}{4} \quad \text{(dividing both sides by 4)}
\]

Now, we can write the second equation \(y = ?\) as any multiple of this equation. For example, if we multiply the entire equation by 2, we get:

\[
y = \frac{2(1 - 3x)}{4} = \frac{2 - 6x}{4} = \frac{1 - 3x}{2}
\]

Thus, the completed statement could be:

The system of equations \(3x + 4y = 1\) and \(y = \frac{1 - 3x}{4}\) has an infinite number of solutions.

Or you could express it with any other scalar multiple:

The system of equations \(3x + 4y = 1\) and \(y = \frac{1 - 3x}{k}\) where \(k\) is any non-zero constant, has an infinite number of solutions.