Problems on Slope and Y-intercept

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.

9th Grade Math

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