Question

Complete the statement with an equation that makes the statement true.(1 point)
The system of equations 3x+4y=1
3
𝑥
+
4
𝑦
=
1
and y=
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 or can be expressed in the same form.

Given the first equation:

\[
3x + 4y = 1
\]

Let's express \( y \) in terms of \( x \) for this equation. To do this, we can solve for \( y \):

1. Rearranging the equation:
\[
4y = 1 - 3x
\]
2. Dividing by 4:
\[
y = \frac{1 - 3x}{4}
\]

Now, to create a second equation that will give us an infinite number of solutions, we could multiply the entire first equation by a non-zero constant \( k \). For example, if we let \( k = 1 \), we would have:

\[
y = \frac{1 - 3x}{4} \Rightarrow \text{(as our second equation)}
\]

Alternatively, we could also write it as:

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

Thus, the complete statement is:

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