Ask a New Question
Menu

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

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 solve the questions, we will analyze the equations provided to identify which form systems with infinite solutions, no solutions, and exactly one solution.

c. Equations that form a system with infinite solutions

For a system to have infinite solutions, the two equations must be identical (essentially, they represent the same line).

From the equations:

  1. y = 5x - 4
  2. y = -2x + 3
  3. y = 3x + 1
  4. y = 3x + 1
  5. y = 5x + 2
  6. y = 4x - 1

The two equations that are identical are:

  • Equation 3: y = 3x + 1
  • Equation 4: y = 3x + 1

So, the answer is: c. y = 3x + 1 and y = 3x + 1

d. Equations that form a system with no solution

For a system to have no solutions, the two lines must be parallel, meaning they have the same slope but different y-intercepts.

Looking at the slopes of the equations:

  • Equation 1: slope = 5
  • Equation 2: slope = -2
  • Equation 3: slope = 3
  • Equation 4: slope = 3
  • Equation 5: slope = 5
  • Equation 6: slope = 4

The equations with the same slope but different y-intercepts are:

  • Equation 1: y = 5x - 4
  • Equation 5: y = 5x + 2

So, the answer is: d. y = 5x - 4 and y = 5x + 2

e. Equations that form a system with one solution

For a system to have exactly one solution, the lines must intersect, which means they have different slopes.

The equations with different slopes are:

  1. y = 5x - 4 (slope = 5)
  2. y = -2x + 3 (slope = -2)

Now, let's find the solution for these two equations by setting them equal to each other:

Set the equations equal: \[ 5x - 4 = -2x + 3 \]

Now, solve for x:

  1. Add \(2x\) to both sides: \[ 5x + 2x - 4 = 3 \] \[ 7x - 4 = 3 \]

  2. Add 4 to both sides: \[ 7x = 7 \]

  3. Divide by 7: \[ x = 1 \]

Now we can substitute \(x = 1\) back into either of the equations to find \(y\). Let's use the first equation: \[ y = 5(1) - 4 \] \[ y = 5 - 4 \] \[ y = 1 \]

The solution to the system is \((1, 1)\).

So, the answer is: e. y = 5x - 4 and y = -2x + 3, solution \((1, 1)\).