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 or a rearrangement of the first equation.

Given the first equation:

\[ 3x + 4y = 1 \]

We can express \(y\) in terms of \(x\). Rearranging the equation gives:

\[ 4y = 1 - 3x \]

Dividing everything by 4:

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

Now, to provide a second equation that results in an infinite number of solutions, we can multiply the first equation by a non-zero constant or simply rearrange the same equation.

For example, we can multiply the entire equation by \(k\), where \(k\) is any non-zero constant. Let's take \(k = 1\) for simplicity.

Thus, the second equation could also be:

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

Or more generally:

\[ 3x + 4y = k(1) \] where \( k \) is any non-zero constant.

By using \(y = \frac{1}{4} - \frac{3}{4}x\) as the second equation, we create a system that has infinite solutions along the line represented by this equation.

Therefore, the statement can be completed as follows:

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