Estimate the solution to the system of these linear equations based on the graph.

y=2x+3
y=12x+6 (1 point)

(7, 2)
(2,−7)

(−2, 7)

(2, 7)
please

1 answer

To estimate the solution to the system of equations given the lines in the equations \(y = 2x + 3\) and \(y = 12x + 6\), we should find the point where these two lines intersect.

  1. Let's first express both equations in standard form.

    • For the first equation, \(y = 2x + 3\): \[ 2x - y + 3 = 0 \]
    • For the second equation, \(y = 12x + 6\): \[ 12x - y + 6 = 0 \]
  2. Next, let's set the equations equal to each other to find the intersection point: \[ 2x + 3 = 12x + 6 \]

  3. Rearranging this gives: \[ 2x - 12x + 3 - 6 = 0 \] \[ -10x - 3 = 0 \] \[ -10x = 3 \] \[ x = -\frac{3}{10} \]

  4. Substitute \(x = -\frac{3}{10}\) back into either of the original equations to find \(y\): \[ y = 2\left(-\frac{3}{10}\right) + 3 = -\frac{6}{10} + 3 = 3 - 0.6 = 2.4 \]

So the intersection point is approximately \((-0.3, 2.4)\).

Now, let's evaluate the provided choices to see if any of them match this estimated point:

  1. (7, 2)
  2. (2, -7)
  3. (-2, 7)
  4. (2, 7)

None of these options match \((-0.3, 2.4)\) closely. However, if we were to consider rounding or estimating based on visual observation from a graph, you might have to choose the one that seems closest based on other observations, but none numerically match the intersection point when calculated.

If you have a graph of these equations available, looking for the estimated point visually would help indicate which of the listed points is nearest to where the lines intersect.