Question

A coordinate grid with 2 lines. The first line is labeled y equals 0.5 x plus 3.5 and passes through (negative 3, 1), (negative 2.7, 2.1), and (0, 3.5). The second line is labeled y equals negative StartFraction 2 over 3 EndFraction x plus StartFraction 1 over 3 EndFraction and passes through the points (negative 4, 3), (negative 2.7, 2.1), and (StartFraction 1 over 3 EndFraction, 0).
Which is the approximate solution to the system y = 0.5x + 3.5 and y = −A system of equations. y equals 0.5 x plus 3.5. y equals negative StartFraction 2 over 3 EndFraction x plus StartFraction 1 over 3 EndFraction.x + shown on the graph?

(–2.7, 2.1)
(–2.1, 2.7)
(2.1, 2.7)
(2.7, 2.1)

Answer 1

To find the approximate solution to the system of equations given by the lines:

1. \( y = 0.5x + 3.5 \)
2. \( y = -\frac{2}{3}x + \frac{1}{3} \)

we need to determine the intersection point of these two lines. This involves setting the two equations equal to each other:

\[
0.5x + 3.5 = -\frac{2}{3}x + \frac{1}{3}
\]

To solve for \( x \), we first eliminate the fractions by multiplying the entire equation by 6 (the least common multiple of the denominators):

\[
6(0.5x) + 6(3.5) = 6\left(-\frac{2}{3}x\right) + 6\left(\frac{1}{3}\right)
\]

This simplifies to:

\[
3x + 21 = -4x + 2
\]

Next, we add \( 4x \) to both sides:

\[
3x + 4x + 21 = 2
\]
\[
7x + 21 = 2
\]

Now, subtract 21 from both sides:

\[
7x = 2 - 21
\]
\[
7x = -19
\]

Finally, dividing each side by 7 gives:

\[
x = -\frac{19}{7} \approx -2.7
\]

Now, we substitute \( x = -\frac{19}{7} \) back into one of the original equations to find \( y \). We can use the first equation \( y = 0.5x + 3.5 \):

\[
y = 0.5\left(-\frac{19}{7}\right) + 3.5
\]
\[
y = -\frac{19}{14} + \frac{49}{14}
\]
\[
y = \frac{30}{14} = \frac{15}{7} \approx 2.14
\]

So the intersection point is approximately \( (-2.7, 2.1) \).

Now, we match this with the options provided:

- (–2.7, 2.1)
- (–2.1, 2.7)
- (2.1, 2.7)
- (2.7, 2.1)

The correct solution is \( (-2.7, 2.1) \).