Question

ear equations and inequalities
which is the best approximation for the solution of the system of equations?
y = -\frac{2}{5}x + 1
y = 3x - 2
(0.88, 1.1)
(0.45, 0.88)
(1.3, 0.88)
(0.88, 0.65)

Explanation:

Step1: Set the equations equal

To find the solution of the system \( y = -\frac{2}{5}x + 1 \) and \( y = 3x - 2 \), set them equal:
\( -\frac{2}{5}x + 1 = 3x - 2 \)

Step2: Solve for \( x \)

Multiply through by 5 to eliminate the fraction:
\( -2x + 5 = 15x - 10 \)
Add \( 2x \) to both sides:
\( 5 = 17x - 10 \)
Add 10 to both sides:
\( 15 = 17x \)
Divide by 17:
\( x = \frac{15}{17} \approx 0.88 \)

Step3: Find \( y \) using \( x \approx 0.88 \)

Substitute \( x \approx 0.88 \) into \( y = 3x - 2 \):
\( y = 3(0.88) - 2 = 2.64 - 2 = 0.64 \approx 0.65 \)? Wait, no—wait, let's check the first equation: \( y = -\frac{2}{5}(0.88) + 1 = -0.352 + 1 = 0.648 \approx 0.65 \). But wait, the options: (0.88, 1.1), (0.45, 0.88), (1.3, 0.88), (0.88, 0.65). Wait, maybe I miscalculated earlier. Wait, no—wait, the blue line is \( y = 3x - 2 \), red is \( y = -\frac{2}{5}x + 1 \). Let's check the intersection. When \( x \approx 0.88 \), \( y \) from \( 3x - 2 \): 3*0.88=2.64, 2.64-2=0.64≈0.65. So the point is (0.88, 0.65)? Wait, but the first option is (0.88, 1.1). Wait, maybe I made a mistake. Wait, no—wait, let's re-solve the system algebraically.

Wait, \( -\frac{2}{5}x + 1 = 3x - 2 \)
\( 1 + 2 = 3x + \frac{2}{5}x \)
\( 3 = \frac{15}{5}x + \frac{2}{5}x = \frac{17}{5}x \)
\( x = 3 * \frac{5}{17} = \frac{15}{17} \approx 0.88 \). Then \( y = 3(\frac{15}{17}) - 2 = \frac{45}{17} - \frac{34}{17} = \frac{11}{17} \approx 0.65 \). So \( y \approx 0.65 \). So the solution is (0.88, 0.65), which is the last option. Wait, but let's check the graph: the intersection is around x≈0.88, y≈0.65 (since the blue line crosses the red line there). The first option (0.88, 1.1) has y=1.1, which is too high. (0.45, 0.88) x too low. (1.3, 0.88) x too high. So (0.88, 0.65) is correct. Wait, but the last option is (0.88, 0.65). Wait, maybe my initial step3 was correct. So the best approximation is (0.88, 0.65).

Answer:

(0.88, 0.65)