Find the slope of the line l in each of the following diagrams.
Step-by-step solution
Idea: Read two clear points from each grid (each small square is 1 unit) and use slope = y2 − y1x2 − x1. A horizontal line has rise 0; a vertical line has run 0, and division by 0 is not defined.
- (i) l is horizontal: take (−2, 3) and (4, 3). Slope = 3 − 34 − (−2) = 06 = 0.1 mark
- (ii) Marked points (−4, 4) and (4, −4). Slope = −4 − 44 − (−4) = −88 = −1.1 mark
- (iii) Marked points (−4, −2) and (2, 2). Slope = 2 − (−2)2 − (−4) = 46 = 23.1 mark
- (iv) l is vertical: take (−2, 1) and (−2, 4). Slope = 4 − 1−2 − (−2) = 30. Division by 0 is not defined, so the slope is undefined.1 mark
Check: (iii) The line crosses the x-axis at (−1, 0): from (−1, 0) to (2, 2) gives 23 again ✓. (ii) The line passes through the origin with y = −x ✓.
Answer to write in the exam
(i) Points (−2, 3), (4, 3): slope = 3 − 34 − (−2) = 06 = 0
(ii) Points (−4, 4), (4, −4): slope = −4 − 44 − (−4) = −88 = −1
(iii) Points (−4, −2), (2, 2): slope = 2 − (−2)2 − (−4) = 46 = 23
(iv) Points (−2, 1), (−2, 4): slope = 4 − 1−2 − (−2) = 30, undefined
∴ Slopes: (i) 0 (ii) −1 (iii) 23 (iv) undefined
Common mistakes that cost marks
- Saying a horizontal line has “no slope” or undefined slope. Its slope is 0; it is the vertical line whose slope is undefined.
- In (ii), giving +1. The line goes down from left to right, so the slope is negative.
- Counting squares from the wrong place: count from the origin (where the axes cross) to each marked point, along the x-axis first and then up or down.
How this can come in the exam
The slope of the line y = −2 is
- −2
- 0
- undefined
- 2
Show answer
(B) 0
y = −2 is horizontal: rise 0, slope 0.
Assertion (A): The line x = 5 has an undefined slope.
Reason (R): For any two points on x = 5, x2 − x1 = 0.
- Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
Show answer
(A) Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
The run is 0 and division by 0 is undefined, so R explains A.
Try one yourself
A line passes through (−3, 5) and (3, 1). Find its slope.
Show answer
1 − 53 − (−3) = −46 = −23.
More questions like this
- An accessibility ramp for wheelchairs is to be made alongside the staircase. The guidelines given for the slope of the ramp is 1 cm vertical rise to 12 cm horizontal length. What should be the horizontal length of the ramp if the total height of the stairs is 18 cm?
- The cost of one paratha and 2 bowls of dahi is ₹90. Also, the cost of 2 parathas and 7 bowls of dahi is ₹260. Can you figure out the cost of one paratha and one bowl of dahi and explain your reasoning?
- Rakesh’s puzzle: “I have thought of two numbers”, he said. “Their sum is 25 and their difference is 11”.
We can also solve this using a pair of linear equations. - Consider another situation. A science exhibition charges an entry fee that is different for adults and children. Here are the amount paid by two different groups.
Group A: 2 adult tickets and 3 child tickets for ₹600
Group B: 3 adult tickets and 2 child tickets for ₹700
How much do individual adult and child tickets cost?
If the cost of one adult ticket is ₹x and one child ticket is ₹y, then: 2x + 3y = 600, 3x + 2y = 700.
This gives us another pair of linear equations in two variables. A solution to these equations will give us the answer that we seek. Can you see why? - Given below are some everyday situations. Construct a pair of linear equations in two variables for each situation:
On two different days a family buys movie tickets and snack boxes. Let the cost of one movie ticket be ₹x and the cost of one snack box be ₹y. Frame a pair of linear equations in x and y to represent the following situations:
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