Question

1. find the distance between the two parallel lines:

$y = -2x + 4$
$y = -2x + 14$

1. find the value of x and y.

$(13y)^circ$
$(8y+75)^circ$
$(2x+34)^circ$

Explanation:

Problem 31: Distance between parallel lines

Step1: Rewrite in standard form

For $y=-2x+4$: $2x + y - 4 = 0$
For $y=-2x+14$: $2x + y - 14 = 0$

Step2: Apply distance formula

Use $d=\frac{|C_1-C_2|}{\sqrt{A^2+B^2}}$ where $Ax+By+C_1=0$, $Ax+By+C_2=0$
$d=\frac{|-4 - (-14)|}{\sqrt{2^2+1^2}}=\frac{|10|}{\sqrt{5}}$

Step3: Simplify the expression

$\frac{10}{\sqrt{5}} = 2\sqrt{5}$

Problem 32: Solve for x and y

Step1: Set up equation for y

Supplementary angles sum to $180^\circ$: $13y + 8y + 75 = 180$

Step2: Solve for y

$21y = 180 - 75 = 105$
$y=\frac{105}{21}=5$

Step3: Set up equation for x

Corresponding angles are equal: $2x + 34 = 13y$

Step4: Substitute y=5, solve for x

$2x + 34 = 13\times5=65$
$2x=65-34=31$
$x=\frac{31}{2}=15.5$

Answer:

1. The distance between the parallel lines is $2\sqrt{5}$
2. $x=15.5$, $y=5$