find the slope of a line parallel to each given line.
13) ( y = -x + 5 )
14) ( x = 3 )

find the slope of the line through each pair of points.
15) ( (14, 13), (4, -1) )
16) ( (13, 14), (10, 2) )

17) ( mangle qrc = 6x + 2 ), ( mangle crs = 85^circ ),
and ( mangle qrs = 21x - 3 ). find ( x ).
18) ( mangle pfe = x + 58 ), ( mangle gfp = x + 103 ),
and ( mangle gfe = 155^circ ). find ( mangle gfp ).

find the measure of the angle indicated in bold.
19) diagram with ( 6x + 7 ) and ( 7x - 1 )
20) diagram with ( 41x + 3 ) and ( 43x - 1 )

Question

find the slope of a line parallel to each given line.
13) ( y = -x + 5 )
14) ( x = 3 )

find the slope of the line through each pair of points.
15) ( (14, 13), (4, -1) )
16) ( (13, 14), (10, 2) )

17) ( mangle qrc = 6x + 2 ), ( mangle crs = 85^circ ),
    and ( mangle qrs = 21x - 3 ). find ( x ).
18) ( mangle pfe = x + 58 ), ( mangle gfp = x + 103 ),
    and ( mangle gfe = 155^circ ). find ( mangle gfp ).

find the measure of the angle indicated in bold.
19) diagram with ( 6x + 7 ) and ( 7x - 1 )
20) diagram with ( 41x + 3 ) and ( 43x - 1 )

Accepted Answer

### Problem 13: Find the slope of a line parallel to $ y = -x + 5 $
# Explanation:
## Step 1: Recall slope-intercept form
The slope-intercept form of a line is $ y = mx + b $, where $ m $ is the slope. For the line $ y = -x + 5 $, the slope $ m $ is $ -1 $.
## Step 2: Parallel lines have equal slopes
Since parallel lines have the same slope, the slope of a line parallel to $ y = -x + 5 $ is also $ -1 $.
# Answer: $ -1 $

### Problem 14: Find the slope of a line parallel to $ x = 3 $
# Explanation:
## Step 1: Analyze the line $ x = 3 $
The line $ x = 3 $ is a vertical line. Vertical lines have an undefined slope.
## Step 2: Parallel lines to vertical lines
Any line parallel to a vertical line is also vertical, so it has an undefined slope.
# Answer: Undefined

### Problem 15: Find the slope of the line through $ (14, 13) $ and $ (4, -1) $
# Explanation:
## Step 1: Recall the slope formula
The slope $ m $ between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is $ m=\frac{y_2 - y_1}{x_2 - x_1} $.
## Step 2: Substitute the points
Let $ (x_1, y_1)=(14, 13) $ and $ (x_2, y_2)=(4, -1) $. Then $ m=\frac{-1 - 13}{4 - 14}=\frac{-14}{-10}=\frac{7}{5} $.
# Answer: $ \frac{7}{5} $

### Problem 16: Find the slope of the line through $ (13, 14) $ and $ (10, 2) $
# Explanation:
## Step 1: Use the slope formula
The slope $ m=\frac{y_2 - y_1}{x_2 - x_1} $ for points $ (x_1, y_1) $ and $ (x_2, y_2) $.
## Step 2: Substitute the values
Let $ (x_1, y_1)=(13, 14) $ and $ (x_2, y_2)=(10, 2) $. Then $ m=\frac{2 - 14}{10 - 13}=\frac{-12}{-3}=4 $.
# Answer: $ 4 $

### Problem 17: Solve for $ x $ given $ m\angle QRC = 6x + 2 $, $ m\angle CRS = 85^\circ $, and $ m\angle QRS = 21x - 3 $
# Explanation:
## Step 1: Angle addition postulate
By the angle addition postulate, $ m\angle QRC + m\angle CRS=m\angle QRS $.
## Step 2: Substitute the expressions
Substitute $ m\angle QRC = 6x + 2 $, $ m\angle CRS = 85^\circ $, and $ m\angle QRS = 21x - 3 $ into the equation: $ 6x + 2+85 = 21x - 3 $.
## Step 3: Simplify and solve for $ x $
Simplify the left side: $ 6x + 87 = 21x - 3 $. Subtract $ 6x $ from both sides: $ 87 = 15x - 3 $. Add 3 to both sides: $ 90 = 15x $. Divide both sides by 15: $ x = 6 $.
# Answer: $ 6 $

### Problem 18: Find $ m\angle GFP $ given $ m\angle PFE = x + 58 $, $ m\angle GFP = x + 103 $, and $ m\angle GFE = 155^\circ $
# Explanation:
## Step 1: Angle addition postulate
By the angle addition postulate, $ m\angle PFE + m\angle GFP=m\angle GFE $.
## Step 2: Substitute the expressions
Substitute $ m\angle PFE = x + 58 $, $ m\angle GFP = x + 103 $, and $ m\angle GFE = 155^\circ $ into the equation: $ (x + 58)+(x + 103)=155 $.
## Step 3: Simplify and solve for $ x $
Simplify the left side: $ 2x + 161 = 155 $. Subtract 161 from both sides: $ 2x=-6 $. Divide by 2: $ x=-3 $.
## Step 4: Find $ m\angle GFP $
Substitute $ x = - 3 $ into $ m\angle GFP = x + 103 $: $ m\angle GFP=-3 + 103 = 100^\circ $.
# Answer: $ 100^\circ $

### Problem 19: Find the measure of the angle indicated ( $ 6x + 7 $ and $ 7x - 1 $ are corresponding angles)
# Explanation:
## Step 1: Corresponding angles are equal
Since the lines are parallel, the corresponding angles $ 6x + 7 $ and $ 7x - 1 $ are equal. So $ 6x + 7 = 7x - 1 $.
## Step 2: Solve for $ x $
Subtract $ 6x $ from both sides: $ 7=x - 1 $. Add 1 to both sides: $ x = 8 $.
## Step 3: Find the angle measure
Substitute $ x = 8 $ into $ 6x + 7 $: $ 6(8)+7 = 48 + 7 = 55^\circ $. (We could also use $ 7x - 1=7(8)-1 = 55^\circ $)
# Answer: $ 55^\circ $

### Problem 20: Find the measure of the angle indicated ( $ 41x + 3 $ and $ 43x - 1 $ are equal as they are corresponding angles)
# Explanation:
## Step 1: Set the angles equal
Since the lines are parallel and the angles are corresponding, $ 41x + 3 = 43x - 1 $.
## Step 2: Solve for $ x $
Subtract $ 41x $ from both sides: $ 3 = 2x - 1 $. Add 1 to both sides: $ 4 = 2x $. Divide by 2: $ x = 2 $.
## Step 3: Find the angle measure
Substitute $ x = 2 $ into $ 41x + 3 $: $ 41(2)+3 = 82 + 3 = 85^\circ $. (We could also use $ 43x - 1=43(2)-1 = 85^\circ $)
# Answer: $ 85^\circ $

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QUESTION IMAGE

Question

Explanation:

Problem 13: Find the slope of a line parallel to \( y = -x + 5 \)

Step 1: Recall slope-intercept form

Step 2: Parallel lines have equal slopes

Step 1: Analyze the line \( x = 3 \)

Step 2: Parallel lines to vertical lines

Step 1: Recall the slope formula

Step 2: Substitute the points

Answer:

Problem 14: Find the slope of a line parallel to \( x = 3 \)