Answer :

To find the best approximation for the solution of the given system of linear equations:
[tex]$ \begin{array}{c} y = -\frac{2}{5} x + 1 \\ y = 3x - 2 \end{array} $[/tex]

we need to determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously. Here is a step-by-step approach to solve this system:

1. Express both equations in terms of [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{2}{5} x + 1 \quad \text{(Equation 1)} \][/tex]
[tex]\[ y = 3x - 2 \quad \text{(Equation 2)} \][/tex]

2. Set the right-hand sides of Equation 1 and Equation 2 equal to each other:
[tex]\[ -\frac{2}{5} x + 1 = 3x - 2 \][/tex]

3. Solve for [tex]\( x \)[/tex]:
- Combine like terms:
[tex]\[ 1 + 2 = 3x + \frac{2}{5} x \][/tex]
[tex]\[ 3 = 3x + \frac{2}{5} x \][/tex]

- Simplify the equation:
[tex]\[ 3 = \left(3 + \frac{2}{5}\right) x \][/tex]
[tex]\[ 3 = \left(\frac{15}{5} + \frac{2}{5}\right) x \][/tex]
[tex]\[ 3 = \frac{17}{5} x \][/tex]

- Isolate [tex]\( x \)[/tex]:
[tex]\[ x = 3 \times \frac{5}{17} \][/tex]
[tex]\[ x = \frac{15}{17} \][/tex]

4. Approximate the value of [tex]\( x \)[/tex]:
[tex]\[ x \approx 0.882 \][/tex]

5. Substitute [tex]\( x \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Using Equation 2:
[tex]\[ y = 3x - 2 \][/tex]
[tex]\[ y = 3 \left(\frac{15}{17}\right) - 2 \][/tex]
[tex]\[ y = \frac{45}{17} - 2 \][/tex]
[tex]\[ y = \frac{45}{17} - \frac{34}{17} \][/tex]
[tex]\[ y = \frac{11}{17} \][/tex]

6. Approximate the value of [tex]\( y \)[/tex]:
[tex]\[ y \approx 0.647 \][/tex]

So, the best approximation for the solution to the system of equations is [tex]\((x, y) \approx (0.882, 0.647)\)[/tex].