Question

1. the equation of line ( h ) is ( 2x + y = 1 ). line ( m ) is the image of line ( h ) after a dilation of scale factor 4 with respect to the origin. what is the equation of the line ( m )?

(1) ( y = -2x + 1 )
(2) ( y = -2x + 4 )
(3) ( y = 2x + 4 )
(4) ( y = 2x + 1 )

Explanation:

Step1: Recall Dilation of a Line

A dilation with respect to the origin with scale factor \( k \) transforms a point \( (x,y) \) to \( (kx,ky) \). For a line \( ax + by = c \), after dilation with scale factor \( k \), the new line will be \( ax+by=kc \) (since if \( (x,y) \) is on the original line, \( (kx,ky) \) should satisfy the new line equation, so \( a(kx)+b(ky)=c' \), which simplifies to \( k(ax + by)=c' \), and since \( ax + by = c \), then \( c'=kc \)).

Step2: Apply Dilation to the Given Line

The equation of line \( l \) is \( 2x + y = 1 \). We apply a dilation with scale factor \( 4 \) with respect to the origin. Using the rule from Step 1, the new equation (for line \( m \)) will be \( 2x + y=4\times1 \), which simplifies to \( 2x + y = 4 \).

Step3: Rewrite in Slope - Intercept Form

We rewrite \( 2x + y = 4 \) in slope - intercept form \( y=mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept). Subtract \( 2x \) from both sides: \( y=-2x + 4 \).

Answer:

(2) \( y = - 2x + 4 \)