To find the approximate solution to the system of linear equations given by [tex]\( y = 0.5x + 3.5 \)[/tex] and [tex]\( y = -\frac{2}{3}x + \frac{1}{3} \)[/tex], we need to determine the point of intersection of these two lines.
### Step-by-Step Solution:
1. Set the equations equal to each other to find the x-coordinate of the intersection:
[tex]\[
0.5x + 3.5 = -\frac{2}{3}x + \frac{1}{3}
\][/tex]
2. Combine like terms by adding [tex]\(\frac{2}{3}x\)[/tex] to both sides:
[tex]\[
0.5x + \frac{2}{3}x + 3.5 = \frac{1}{3}
\][/tex]
3. Convert 0.5 to a fraction to make the arithmetic easier:
[tex]\[
\frac{1}{2}x + \frac{2}{3}x = \frac{1}{3} - 3.5
\][/tex]
4. Find a common denominator for the fractions:
[tex]\[
\frac{3}{6}x + \frac{4}{6}x = \frac{1}{3} - \frac{10.5}{3}
\][/tex]
Simplify the left side:
[tex]\[
\frac{7}{6}x = \frac{1}{3} - \frac{10.5}{3}
\][/tex]
5. Simplify the right side:
[tex]\[
\frac{1}{3} - \frac{10.5}{3} = \frac{1 - 10.5}{3} = \frac{-9.5}{3}
\][/tex]
6. Express [tex]\(\frac{-9.5}{3}\)[/tex] as a single fraction:
[tex]\[
\frac{7}{6}x = \frac{-9.5}{3}
\][/tex]
7. Solve for [tex]\( x \)[/tex] by multiplying both sides by the reciprocal of [tex]\(\frac{7}{6}\)[/tex]:
[tex]\[
x = \frac{-9.5}{3} \times \frac{6}{7}
\][/tex]
Simplify this:
[tex]\[
x = \frac{-9.5 \times 6}{3 \times 7} = \frac{-57}{21} = -\frac{57}{21} = -2.7
\][/tex]
8. Substitute [tex]\( x = -2.7 \)[/tex] back into one of the original equations to find the y-coordinate:
Use [tex]\( y = 0.5x + 3.5 \)[/tex]:
[tex]\[
y = 0.5 \times (-2.7) + 3.5
\][/tex]
Simplify this:
[tex]\[
y = -1.35 + 3.5 = 2.15
\][/tex]
Approximate to one decimal place:
[tex]\[
y \approx 2.1
\][/tex]
### Conclusion
The approximate solution to the system is [tex]\((-2.7, 2.1)\)[/tex].
### Therefore, the answer is: [tex]\((-2.7, 2.1)\)[/tex].
