Question

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

Answer 1

so what is the answer

Answer 2

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 that both equations represent the same line.

Starting with the first equation:

\[
3x + 4y = 1
\]

We can express \(y\) in terms of \(x\):

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

Now, we can express this in the form of \(y = mx + b\):

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

Now to generate an equation that has the same slope and the same intercept, we can multiply the entire equation by any non-zero constant \(k\):

Let's choose \(k = 2\):

\[
2(3x + 4y) = 2(1) \implies 6x + 8y = 2
\]

Thus, we can write:

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

So the equation can be presented in the form \(y = -\frac{3}{4}x + \frac{1}{4}\).

The complete statement can be:

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

Answer 3

To make the statement true, you can complete it with the following equation:

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

Alternatively, you can also express the second equation as:

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

Both forms achieve the requirement that the system has an infinite number of solutions since the second equation is equivalent to the first.