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.

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

1. Rewrite the original equation:
\( 4y = 1 - 3x \)

2. Divide everything by 4:
\( y = \frac{1}{4} - \frac{3}{4}x \)

Now, say we multiply the entire equation by any non-zero constant \( k \).

For example, let \( k = 2 \):
\[
y = 2 \left(\frac{1}{4} - \frac{3}{4}x\right) = \frac{1}{2} - \frac{3}{2}x
\]

The equation that results is:
\[
y = -\frac{3}{2}x + \frac{1}{2}
\]

Thus, completing the statement:
The system of equations 3x + 4y = 1 and \( y = -\frac{3}{2}x + \frac{1}{2} \) has an infinite number of solutions.

You could also choose other multiples, but they all must reflect the linear relationship derived from the original equation.