Question

1. in the standard $(z,y)$ coordinate plane, what is the equation of the line that passes through the origin and the point $(3,4)$?

a. $y=\frac{1}{4}x+\frac{3}{4}$
b. $y=\frac{1}{4}x - \frac{1}{3}$
c. $y=\frac{4}{3}x$
d. $y=\frac{1}{2}x+\frac{3}{4}$
e. $y=\frac{9}{4}x$

1. the minimum fine for driving in excess of the speed limit is $25. an additional $6 is added to the minimum fine for each mile per hour (mph) in excess of the speed limit. rachel was issued a $103 fine for speeding in a 55 - mph speed limit zone. for driving at what speed, in mph, was rachel fined?

a. 48
b. 52
c. 62
d. 68
e. 78

Explanation:

Step1: Find the slope of the line

The formula for the slope \(m\) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Since the line passes through the origin \((0,0)\) and the point \((3,4)\), then \(x_1 = 0,y_1=0,x_2 = 3,y_2 = 4\).
So \(m=\frac{4-0}{3 - 0}=\frac{4}{3}\).

Step2: Write the equation of the line

The equation of a line in slope - intercept form is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.
Since the line passes through the origin \((0,0)\), when \(x = 0,y=0\), substituting into \(y=mx + b\) gives \(0=m\times0 + b\), so \(b = 0\).
The equation of the line is \(y=\frac{4}{3}x\).

Answer:

C. \(y=\frac{4}{3}x\)