Question

Which ordered pair is the best estimate for the solution of the system of equations?

y=3/2x+6 y=1/4x−2



Responses

(−7, −4)
Begin ordered pair negative 7 comma negative 4 end ordered pair.

(−7, −3.5)
Begin ordered pair negative 7 comma negative 3.5 end ordered pair.

(−6, −3)
Begin ordered pair negative 6 comma negative 3 end ordered pair.

(−6.5, −3.5)
Begin ordered pair negative 6.5 comma negative 3.5 end ordered pair.

Answer 1

To estimate the solution of the system of equations given by:

1. \( y = \frac{3}{2}x + 6 \)
2. \( y = \frac{1}{4}x - 2 \)

We can find the intersection of these two lines by setting them equal to each other:

\[
\frac{3}{2}x + 6 = \frac{1}{4}x - 2
\]

To eliminate the fractions, we can multiply every term by 4:

\[
4 \left( \frac{3}{2}x \right) + 4(6) = 4 \left( \frac{1}{4}x \right) - 4(2)
\]

This simplifies to:

\[
6x + 24 = x - 8
\]

Next, we can rearrange the terms:

\[
6x - x = -8 - 24
\]

\[
5x = -32
\]

Dividing both sides by 5 gives:

\[
x = -\frac{32}{5} = -6.4
\]

Now, we can substitute \( x = -6.4 \) back into either equation to find \( y \). Using the first equation:

\[
y = \frac{3}{2}(-6.4) + 6
\]
\[
y = -9.6 + 6
\]
\[
y = -3.6
\]

Thus, the estimated solution of the system is approximately \( (-6.4, -3.6) \).

Now, we can compare this to the provided options:

1. \( (-7, -4) \)
2. \( (-7, -3.5) \)
3. \( (-6, -3) \)
4. \( (-6.5, -3.5) \)

The closest option to \( (-6.4, -3.6) \) is **(−6.5, −3.5)**.

Thus, the best estimate for the solution of the system of equations is:

**(−6.5, −3.5)**.