Here we will learn how to solve different types of problems on slope and y-intercept.
1. (i) Determine the slope and y-intercept of the line 4x + 7y + 5 = 0
Solution:
Here, 4x + 7y + 5 = 0
⟹ 7y = -4x – 5
⟹ y = -\(\frac{4}{7}\)x - \(\frac{5}{7}\).
Comparing this with y = mx + c, we have: m = -\(\frac{4}{7}\) and c = - \(\frac{5}{7}\)
Therefore, slope = -\(\frac{4}{7}\) and y-intercept = - \(\frac{5}{7}\)
(ii) Determine the slope and y-intercept of the line 9x - 5y + 2 = 0
Solution:
Here, 9x - 5y - 2 = 0
⟹ -5y = -9x + 2
⟹ y = \(\frac{-9}{-5}\)x + \(\frac{2}{-5}\).
⟹ y = \(\frac{9}{5}\)x - \(\frac{2}{5}\).
Comparing this with y = mx + c, we have: m = \(\frac{9}{5}\) and c = -\(\frac{2}{5}\)
Therefore, slope = \(\frac{9}{5}\) and y-intercept = -\(\frac{2}{5}\)
(iii) Determine the slope and y-intercept of the line 9y + 4 = 0
Solution:
Here, 9y + 4 = 0
⟹ 9y = -4
⟹ y = -\(\frac{4}{9}\)
⟹ y = 0 ∙ x -\(\frac{4}{9}\)
Comparing this with y = mx + c, we have: m = 0 and c = \(\frac{-4}{9}\)
Therefore, slope = 0 and y-intercept = \(\frac{-4}{9}\)
2. The points (-2, 5) and (1, -4) are plotted in the x-y plane. Find the slope and y-intercept of the line joining the points.
Solution:
Let the line graph obtained by joining the points (-2, 5) and (1, -4) be the graph of y = mx + c. So, the given pairs of values of (x, y) obey the relation y = mx + c.
Therefore, 5 = -2m + c ................................. (i)
-4 = m + c ................................. (ii)
Subtracting (ii) from (i), we get:
5 + 4 = -2m – m
⟹ 9 = -3m
⟹ -3m = 9
⟹ m = \(\frac{9}{-3}\)
⟹ m = -3
Putting m = -3 in (ii), we have: -4 = -3 + c
⟹ c = -1.
Now, m = -3 ⟹ the slope of the line graph = -3,
c = -1 ⟹ the y-intercept of the line graph = -1.
On drawing the graph of y = mx + c using slope and y-intercept.
3. Draw the graph of 3x - √3y = 2√3 using its slope and y-intercept.
Solution:
Here, 3x - √3y = 2√3
⟹ - √3y = -3x + 2√3
⟹ √3y = 3x - 2√3
y = √3x – 2
Comparing with y = mx + c, we find the slope m = √3 and y-intercept = -2.
Now, m = tan θ = √3
⟹ θ = 60°.
So, the graph is as shown on the above figure.
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