Each set of ordered pairs represents two points on a line. Match each set of ordered pairs with the description of the line they are on.
Tiles
( [tex]$-5,-3$[/tex] ) and ( [tex]$-5,3$[/tex] )
( [tex]$-7,-1$[/tex] ) and ( [tex]$-1,-7$[/tex] )
( [tex]$-6,-2$[/tex] ) and ( [tex]$6,-2$[/tex] )
Pairs
vertical line $\square$
neither horizontal nor vertical $\square$
horizontal line $\square$
Answer (1)
Calculate the slope for each set of points using the formula m = x 2 − x 1 y 2 − y 1 .
If the slope is undefined (division by zero), the line is vertical.
If the slope is 0, the line is horizontal.
If the slope is neither undefined nor 0, the line is neither horizontal nor vertical. The matches are: ( − 5 , − 3 ) and ( − 5 , 3 ) → vertical line; ( − 7 , − 1 ) and ( − 1 , − 7 ) → neither horizontal nor vertical; ( − 6 , − 2 ) and ( 6 , − 2 ) → horizontal line.
Explanation
Problem Analysis We are given three sets of ordered pairs and we need to match each set with the description of the line they are on: vertical, horizontal, or neither. To do this, we will calculate the slope of the line passing through each pair of points.
Slope Formula The slope, m , of a line passing through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 If the denominator is zero, the slope is undefined, which means the line is vertical. If the numerator is zero, the slope is zero, which means the line is horizontal.
First Set of Points For the first set of points ( − 5 , − 3 ) and ( − 5 , 3 ) , the slope is: m = − 5 − ( − 5 ) 3 − ( − 3 ) = 0 6 Since the denominator is zero, the slope is undefined. This means the line is vertical.
Second Set of Points For the second set of points ( − 7 , − 1 ) and ( − 1 , − 7 ) , the slope is: m = − 1 − ( − 7 ) − 7 − ( − 1 ) = 6 − 6 = − 1 Since the slope is − 1 , the line is neither horizontal nor vertical.
Third Set of Points For the third set of points ( − 6 , − 2 ) and ( 6 , − 2 ) , the slope is: m = 6 − ( − 6 ) − 2 − ( − 2 ) = 12 0 = 0 Since the slope is 0 , the line is horizontal.
Final Answer Therefore, the matches are:
( − 5 , − 3 ) and ( − 5 , 3 ) → vertical line
( − 7 , − 1 ) and ( − 1 , − 7 ) → neither horizontal nor vertical
( − 6 , − 2 ) and ( 6 , − 2 ) → horizontal line
Examples
Understanding the slope of a line is crucial in many real-world applications. For instance, engineers use slopes to design roads and bridges, ensuring they are not too steep. Architects use slopes to design roofs for proper water runoff. In economics, the slope of a supply or demand curve can tell you how responsive the quantity is to changes in price. Knowing how to calculate and interpret slopes helps in making informed decisions in various fields.
