Question

1. solve the system of equations. how many solutions does the system have?

6x + 3y = 24
y = -2x + 8
a no solution
b one solution: x = 2, y = 4
c one solution: x = 3, y = 2
d infinitely many solutions

1. taxi a charges a fee of $3.50, plus $1.75 per mile. taxi b charges a fee of $1.25, plus $2.00 per mile. at what distance would the taxis cost the same?

a 2 miles
b 5 miles
c 9 miles
d 15 miles

(image of two lines on a grid with options: a no solution, b one solution: x=0, y=0, c one solution: x=0, y=5, d infinitely many solutions)

Explanation:

Question 2 (Graph - System of Equations)

Step1: Analyze the graph

The two lines in the graph are parallel (they never intersect). For a system of linear equations represented by lines, parallel lines mean no solution (since they never meet, so no common point).

Step2: Match with options

Option A says "No solution", which matches the analysis of parallel lines.

Step1: Define variables and equations

Let \( d \) be the distance in miles. Cost for Taxi A: \( C_A = 3.50 + 1.75d \). Cost for Taxi B: \( C_B = 1.25 + 2.00d \). Set \( C_A = C_B \): \( 3.50 + 1.75d = 1.25 + 2.00d \).

Step2: Solve for \( d \)

Subtract \( 1.75d \) and \( 1.25 \) from both sides: \( 3.50 - 1.25 = 2.00d - 1.75d \). So, \( 2.25 = 0.25d \). Divide both sides by \( 0.25 \): \( d=\frac{2.25}{0.25}=9 \)? Wait, no, wait: Wait, \( 3.50 - 1.25 = 2.25 \), \( 2.00d - 1.75d = 0.25d \). So \( d = 2.25 / 0.25 = 9 \)? But wait, let's re - calculate: Wait, \( 3.50 - 1.25 = 2.25 \), \( 2.00 - 1.75 = 0.25 \). So \( d=2.25\div0.25 = 9 \)? But the options have 9 as option C? Wait, no, wait, maybe I made a mistake. Wait, Taxi A: \( 3.50+1.75d \), Taxi B: \( 1.25 + 2.00d \). Set equal: \( 3.50 - 1.25=2.00d - 1.75d \), \( 2.25 = 0.25d \), \( d = 9 \). But the options: Wait, the options are A:2, B:5, C:9, D:15. So the correct answer is C? Wait, but let's check with \( d = 9 \): Taxi A: \( 3.50+1.75\times9=3.50 + 15.75 = 19.25 \). Taxi B: \( 1.25+2.00\times9=1.25 + 18 = 19.25 \). Yes, so \( d = 9 \) miles.

Step1: Simplify the first equation

Given \( 6x + 3y = 24 \), divide both sides by 3: \( 2x + y = 8 \). Then, solve for \( y \): \( y=-2x + 8 \).

Step2: Compare with the second equation

The second equation is \( y=-2x + 8 \). So both equations are the same line (they are equivalent). For a system of linear equations, if the two equations represent the same line, there are infinitely many solutions (all points on the line are solutions).

Answer:

A. No solution

Question 3 (Taxi Cost Problem)