Question

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:

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

we need to find the point where these two lines intersect.

**Step 1:** Set the equations equal to each other to find \(x\):

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

**Step 2:** To eliminate the fractions, let's multiply through 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:

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

**Step 3:** Now, combine the like terms:

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

\[
7x = -19
\]

\[
x = -\frac{19}{7} \approx -2.71
\]

**Step 4:** Substitute \(x\) back into one of the original equations to find \(y\). We'll use the first equation:

\[
y = 0.5(-\frac{19}{7}) + 3.5
\]

Calculate \(0.5 \times -\frac{19}{7}\):

\[
y = -\frac{19}{14} + \frac{49}{14} = \frac{30}{14} = \frac{15}{7} \approx 2.14
\]

So, the approximate intersection point is:

\[
\left(-2.71, 2.14\right)
\]

Given the options:

\((-2.7, 2.1)\)

\((-2.1, 2.7)\)

\((2.1, 2.7)\)

\((2.7, 2.1)\)

The closest match is:

\(\boxed{(-2.7, 2.1)}\)