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y=2x+3

y=12x+6

(1 point)
Responses

(2,7)
(
2
,
7
)
open paren 2 comma 7 close paren

(2,−7)
(
2
,
−
7
)
open paren 2 comma negative 7 close paren

(7,2)
(
7
,
2
)
open paren 7 comma 2 close paren

(−2,7 )

3 answers

To find the intersection of the two equations \( y = 2x + 3 \) and \( y = 12x + 6 \), we can set the two equations equal to each other since they both equal \( y \):

\[ 2x + 3 = 12x + 6 \]

Now, let's solve for \( x \):

  1. Subtract \( 2x \) from both sides: \[ 3 = 10x + 6 \]
  2. Subtract 6 from both sides: \[ -3 = 10x \]
  3. Divide by 10: \[ x = -\frac{3}{10} \]

Now we find the corresponding \( y \) value by substituting \( x \) back into either of the original equations. We'll use the first equation \( y = 2x + 3 \):

\[ y = 2\left(-\frac{3}{10}\right) + 3 \] \[ y = -\frac{6}{10} + 3 \] \[ y = -\frac{6}{10} + \frac{30}{10} \] \[ y = \frac{24}{10} = \frac{12}{5} \]

Thus, the point of intersection is \( \left(-\frac{3}{10}, \frac{12}{5}\right) \).

None of the provided response options \( (2, 7) \), \( (2, -7) \), \( (7, 2) \), \( (-2, 7) \) match the intersection point. Therefore, please verify the equations or the options given.

really? i thought it was 2, 7

Let’s check to see if \( (2, 7) \) is indeed a solution for the equations \( y = 2x + 3 \) and \( y = 12x + 6 \).

  1. Substituting \( x = 2 \) into the first equation: \[ y = 2(2) + 3 = 4 + 3 = 7 \]

    So, the point \( (2, 7) \) satisfies the first equation.

  2. Now, substituting \( x = 2 \) into the second equation: \[ y = 12(2) + 6 = 24 + 6 = 30 \]

    The point \( (2, 7) \) does not satisfy the second equation, which gives a \( y \) value of 30 when \( x = 2 \).

Thus, \( (2, 7) \) is not a solution to the system of equations since it does not satisfy both.

If we look further, we can analyze the original equations to find the correct intersection point. I previously calculated that the intersection point is \( \left(-\frac{3}{10}, \frac{12}{5}\right) \), not \( (2, 7) \).

If you need further clarification or if you believe there could be an error elsewhere, please let me know!