Question

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

To create a system of equations that would make the statement true, we need to express \( y \) in terms of \( x \) based on the equation \( 3x + 4y = 1 \) and then derive another equation that is equivalent to it.

First, let's express \( y \) in terms of \( x \) from the given equation \( 3x + 4y = 1 \):

1. Isolate \( 4y \):
\[
4y = 1 - 3x
\]
2. Solve for \( y \):
\[
y = \frac{1 - 3x}{4}
\]

Now, any equation that is a multiple of this equation will also represent the same line and therefore will have an infinite number of solutions when combined with \( 3x + 4y = 1 \).

One simple form of \( y \) that would work is a scalar multiple of the previous equation. For example, if we multiply the entire equation for \( y \) by 1, we still have:

\[
y = \frac{1 - 3x}{4}
\]

Or, alternatively, we can express \( y \) as follows:

\[
y = -\frac{3}{4}x + \frac{1}{4}
\]

Thus, the completed statement could be:

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