YOU MUST USE ALL 6 EQUATIONS from "a" to answer c,d, and e. You cannot use an equation more than once. c. What two equations from those above would form a system with infinite solutions. You can list the 2 equations here: and d. What two equations from those above would form a system with no solution. You can list the 2 equations here: and e. What two equations from those above would form a system with one solution. AND give the solution (x,y) for both equations (x, y) HERE: ( ) You can list the 2 equations here: and MUST SHOW WORK ON HOW YOU GOT YOUR SOLUTION

the equations are:
1. y = 5x - 4
2. y = - 2x + 3
3. y = 3x + 1
4. y = 3x + 1
5. y = 5x + 2
6. y = 4x - 1

1 answer

To determine which pairs of equations from the set form systems with infinite solutions, no solution, and one solution, we can analyze the equations.

Here are the equations:

\( y = 5x - 4 \)
\( y = -2x + 3 \)
\( y = 3x + 1 \)
\( y = 3x + 1 \) (This is a duplicate of equation 3 and will not be used.)
\( y = 5x + 2 \)
\( y = 4x - 1 \)

Two equations will have infinite solutions if they are equivalent (i.e., they are the same line).

Thus, the pair is:

Two equations will have no solution if they have the same slope but different y-intercepts.

The equations are:
- \( y = 5x - 4 \) (slope = 5)
- \( y = 5x + 2 \) (slope = 5, different y-intercept)

Thus, the pair is:

Two equations will have one solution if they have different slopes.

The equations are:
- \( y = 5x - 4 \) (slope = 5)
- \( y = -2x + 3 \) (slope = -2)

To find the solution \((x,y)\), we set these two equations equal to each other:

Set the equations equal: \[ 5x - 4 = -2x + 3 \]
Rearrange to isolate \(x\): \[ 5x + 2x = 3 + 4 \ 7x = 7 \ x = 1 \]
Substitute \(x = 1\) back into either equation to find \(y\): Using \(y = 5x - 4\): \[ y = 5(1) - 4 = 5 - 4 = 1 \]

Thus, the solution is:

To summarize: