Match the line descriptions with the equations.
$y=0$
$9x+3y=18$
the line through $(-5,7)$ and $(-5,-8)$
the line through $(4,-1)$ and $(8,7)$
Possible answers:
0
2
undefined
Answer (1)
The line y = 0 is a horizontal line, so its slope is 0.
The line 9 x + 3 y = 18 can be rewritten as y = − 3 x + 6 , so its slope is -3.
The line through ( − 5 , 7 ) and ( − 5 , − 8 ) is a vertical line, so its slope is undefined.
The line through ( 4 , − 1 ) and ( 8 , 7 ) has slope 8 − 4 7 − ( − 1 ) = 4 8 = 2 .
The final answer is:
y = 0 : 0
9 x + 3 y = 18 : -3
the line through ( − 5 , 7 ) and ( − 5 , − 8 ) : undefined
the line through ( 4 , − 1 ) and ( 8 , 7 ) : 2
Explanation
Understanding the Problem We are given four lines and asked to match them to their slopes. The possible slopes are 0, 2, and undefined.
Finding the Slope of y = 0 The first line is given by the equation y = 0 . This is a horizontal line, and horizontal lines have a slope of 0.
Finding the Slope of 9x + 3y = 18 The second line is given by the equation 9 x + 3 y = 18 . To find the slope, we can rewrite this equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Subtracting 9 x from both sides gives 3 y = − 9 x + 18 . Dividing both sides by 3 gives y = − 3 x + 6 . Therefore, the slope of this line is -3.
Finding the Slope of the Line Through (-5, 7) and (-5, -8) The third line passes through the points ( − 5 , 7 ) and ( − 5 , − 8 ) . Using the slope formula, m = x 2 − x 1 y 2 − y 1 , we have m = − 5 − ( − 5 ) − 8 − 7 = 0 − 15 . Since division by zero is undefined, the slope of this line is undefined.
Finding the Slope of the Line Through (4, -1) and (8, 7) The fourth line passes through the points ( 4 , − 1 ) and ( 8 , 7 ) . Using the slope formula, m = x 2 − x 1 y 2 − y 1 , we have m = 8 − 4 7 − ( − 1 ) = 4 8 = 2 . Therefore, the slope of this line is 2.
Matching Lines to Slopes Matching each line to its corresponding slope:
y = 0 has a slope of 0.
9 x + 3 y = 18 has a slope of -3.
The line through ( − 5 , 7 ) and ( − 5 , − 8 ) has an undefined slope.
The line through ( 4 , − 1 ) and ( 8 , 7 ) has a slope of 2.
Final Answer The slopes are:
y = 0 : 0
9 x + 3 y = 18 : -3
the line through ( − 5 , 7 ) and ( − 5 , − 8 ) : undefined
the line through ( 4 , − 1 ) and ( 8 , 7 ) : 2
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, in construction, the slope of a ramp determines its steepness and accessibility. In economics, the slope of a supply or demand curve indicates how sensitive the quantity supplied or demanded is to changes in price. In navigation, the slope of a path represents the rate of change in altitude or depth. By mastering the concept of slope, you can analyze and solve problems in various fields, making informed decisions and predictions.
