To find the solution for the system of linear equations:
[tex]\[
\begin{cases}
y = -\frac{2}{5}x + 1 \\
y = 3x - 2
\end{cases}
\][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] where both equations intersect. Here is a detailed step-by-step solution:
1. Set the two equations equal to each other: Since both equations are equal to [tex]\( y \)[/tex], we can set them equal to each other to find [tex]\( x \)[/tex]:
[tex]\[
-\frac{2}{5} x + 1 = 3x - 2
\][/tex]
2. Eliminate the fractions: To simplify the equation, eliminate the fraction by multiplying every term by 5:
[tex]\[
-2x + 5 = 15x - 10
\][/tex]
3. Combine like terms: Add [tex]\( 2x \)[/tex] to both sides to start isolating [tex]\( x \)[/tex]:
[tex]\[
5 = 17x - 10
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Add 10 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
15 = 17x
\][/tex]
Now, divide both sides by 17:
[tex]\[
x = \frac{15}{17} \approx 0.8824
\][/tex]
5. Substitute [tex]\( x \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]: Let's use the second equation [tex]\( y = 3x - 2 \)[/tex]:
[tex]\[
y = 3\left(\frac{15}{17}\right) - 2
\][/tex]
Multiply [tex]\( 3 \)[/tex] by [tex]\( \frac{15}{17} \)[/tex]:
[tex]\[
y = \frac{45}{17} - 2
\][/tex]
Convert 2 to a fraction with denominator 17:
[tex]\[
y = \frac{45}{17} - \frac{34}{17}
\][/tex]
Subtract the fractions:
[tex]\[
y = \frac{45 - 34}{17} = \frac{11}{17} \approx 0.6471
\][/tex]
The best approximation for the solution of the system of equations is:
[tex]\[
x \approx 0.8824, \quad y \approx 0.6471
\][/tex]
Therefore, the intersection point of the two lines is approximately [tex]\((0.8824, 0.6471)\)[/tex].
