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].
