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Susie determined that the system of equations below has infinitely many solutions.

The first line goes through the points (2, -13) and (-2, -1).

The second line goes through the points (5, -22) and (-3, 2).

Is she correct? *Use your Desmos calculator

(1 point)
Responses

No, the lines are parallel.
No, the lines are parallel.

Yes, Susie is correct.
Yes, Susie is correct.

No, there is no point of intersection.

She may or may not be correct as the lines possibly cross.

5 answers

To determine if Susie is correct in saying that the system of equations has infinitely many solutions, we first need to find the equations of the two lines described by the points.

  1. First Line (through (2, -13) and (-2, -1)):

    • The slope \( m_1 \) is calculated as: \[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-13)}{-2 - 2} = \frac{12}{-4} = -3 \]
    • Using the point-slope form \( y - y_1 = m(x - x_1) \) with point (2, -13): \[ y + 13 = -3(x - 2) \implies y + 13 = -3x + 6 \implies y = -3x - 7 \]

  2. Second Line (through (5, -22) and (-3, 2)):

    • The slope \( m_2 \) is calculated as: \[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-22)}{-3 - 5} = \frac{24}{-8} = -3 \]
    • Using the point-slope form \( y - y_1 = m(x - x_1) \) with point (5, -22): \[ y + 22 = -3(x - 5) \implies y + 22 = -3x + 15 \implies y = -3x - 7 \]

Both lines are represented by the equation \( y = -3x - 7 \). Since they are the same line, they indeed have infinitely many solutions.

Therefore, the correct response is:

Yes, Susie is correct.

The graph shows how the values of two used cars changed during a 10 year period.

(2 points)
After (Response area) years, the cars both have a value of $(Response area.)

To answer your question about the values of the two used cars over a 10-year period, I'd need to see the graph you mentioned. However, since I cannot view images or graphs, I will guide you on how to extract the information from the graph.

  1. Identify the Intersection Point: Look for the point where the two lines representing the values of the cars intersect. This point will indicate the number of years after which both cars have the same value.

  2. Determine the Value: At the intersection point, trace down to the y-axis to find out what the corresponding value of the cars is at that year.

  3. Complete the Response: You should be able to fill in the following:

    • For the number of years: “After (number of years) years”
    • For the value: “the cars both have a value of $(value)”

If you can provide the values or describe the graph, I would be happy to help you interpret it further!

Put the steps in order that are used to solve the following systems of equations by substitution.

−7x−2y=−13


x−2y=11


(10 points)

responses
x - 2 (-4) = 11
x = 3
-16y - 77 = -13
y = -4
(3, -4
x + 8 = 11
-7 (2y + 11) - 2y = -13
x - 2y = 11 --> x = 2y + 11
-16y = 64
-14y - 77 - 2y = -13

To solve the system of equations by substitution, we will follow a logical sequence of steps. The two equations are:

  1. \(-7x - 2y = -13\) (Equation 1)
  2. \(x - 2y = 11\) (Equation 2)

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. Here are the steps in the correct order:

  1. Isolate \(x\) in Equation 2: \[ x - 2y = 11 \quad \Rightarrow \quad x = 2y + 11 \]

  2. Substitute \(x\) into Equation 1: \[ -7(2y + 11) - 2y = -13 \]

  3. Distribute and simplify: \[ -14y - 77 - 2y = -13 \quad \Rightarrow \quad -16y - 77 = -13 \]

  4. Solve for \(y\): \[ -16y = 64 \quad \Rightarrow \quad y = -4 \]

  5. Substitute \(y\) back into the expression for \(x\): \[ x = 2(-4) + 11 \quad \Rightarrow \quad x = -8 + 11 \quad \Rightarrow \quad x = 3 \]

  6. Final solution: \[ (3, -4) \]

So, the order of steps is:

  1. \(x - 2y = 11 \rightarrow x = 2y + 11\)
  2. \(-7(2y + 11) - 2y = -13\)
  3. \(-14y - 77 - 2y = -13\)
  4. \(-16y = 64\)
  5. \(y = -4\)
  6. \(x = 3\)
  7. \((3, -4)\)

If you match these sequences to the provided response options, it should be easy to align them according to their logical order. Let me know if you need further assistance!