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Which of the following uses the substitution method to solve linear-quadratic system y=x^2−4x+8, y=3x−4

Responses

(4,8)and (3,5)

(4,8)

(−4,−16) and (−3,−13)

(3,5)

3 answers

To solve the linear-quadratic system using the substitution method, we have the equations:

  1. \( y = x^2 - 4x + 8 \)
  2. \( y = 3x - 4 \)

We can set these equations equal to each other since they both equal \( y \):

\[ x^2 - 4x + 8 = 3x - 4 \]

Now, rearranging the equation:

\[ x^2 - 4x - 3x + 8 + 4 = 0 \]

This simplifies to:

\[ x^2 - 7x + 12 = 0 \]

Next, we can factor this quadratic equation:

\[ (x - 3)(x - 4) = 0 \]

This gives us the solutions:

\[ x = 3 \quad \text{or} \quad x = 4 \]

Now we can find the corresponding \( y \) values using the linear equation \( y = 3x - 4 \).

  1. For \( x = 3 \): \[ y = 3(3) - 4 = 9 - 4 = 5. \] So one solution is \( (3, 5) \).

  2. For \( x = 4 \): \[ y = 3(4) - 4 = 12 - 4 = 8. \] So the other solution is \( (4, 8) \).

Thus, the solutions to the system are \( (3, 5) \) and \( (4, 8) \).

From the options provided, the correct response is:

(4,8) and (3,5).

Use the substitution method to find the solutions to the linear-quadratic system y=x^2−x−40, y=−3x−5

Responses

(5,−10)

(5,-20)and (−7,16)

(−7,26)

(5,−10) and (−7,26)

To find the solutions to the system of equations using the substitution method, we have:

  1. \( y = x^2 - x - 40 \)
  2. \( y = -3x - 5 \)

We will set these two expressions for \( y \) equal to each other:

\[ x^2 - x - 40 = -3x - 5 \]

Next, we'll rearrange this equation to set it to zero:

\[ x^2 - x + 3x - 40 + 5 = 0 \]

This simplifies to:

\[ x^2 + 2x - 35 = 0 \]

Now, we can factor this quadratic equation:

\[ (x + 7)(x - 5) = 0 \]

From this factorization, we get the solutions:

\[ x = -7 \quad \text{or} \quad x = 5 \]

Next, we find the corresponding \( y \) values using the second equation \( y = -3x - 5 \):

  1. For \( x = 5 \): \[ y = -3(5) - 5 = -15 - 5 = -20. \] So one solution is \( (5, -20) \).

  2. For \( x = -7 \): \[ y = -3(-7) - 5 = 21 - 5 = 16. \] So the other solution is \( (-7, 16) \).

Thus, the solutions to the system are \( (5, -20) \) and \( (-7, 16) \).

From the options provided, the correct response is:

(5, -20) and (−7, 16).