FINDING X AND Y INTERCEPTS AND FINDING EQUATION FROM INTERCEPTS

Question 1 :

Find the equation of the straight line whose x and y-intercepts on the axes are given by 

(i) 2 and 3

(ii) -1/3 and 3/2

(iii) 2/5 and -3/4

Solution :

(i) 2 and 3

We can find the equation of the line using x and y intercepts, we can use the formula given below. 

(x/a) + (y/b) = 1

here "a" stands for x-intercept and "b" stands for y-intercept.

(x/2) + (y/3)  =  1

(3x + 2y)/6  =  1

3x + 2y  =  6 (or) 3x + 2y - 6  =  0 

(ii) -1/3 and 3/2

We can find the equation of the line using x and y intercepts, we can use the formula given below. 

(x/a) + (y/b)  =  1

here "a" stands for x-intercept and "b" stands for y-intercept.

x/(-1/3) + y/(3/2)  =  1

(-3x/1) + (2y/3) = 1

(-9x + 2y)/3 = 1 

 -9x + 2y = 3 

-9x + 2y - 3 = 0

9x - 2y + 3 = 0

Hence the required equation of the line is 9x - 2y + 3 = 0.

(iii) 2/5 and -3/4

a = 2/5 and b = -3/4

x/(2/5) + y/(-3/4)  =  1

(5x/2) - (4y/3) = 1

(15x - 8y)/6 = 1 

15 x - 8 y = 6

15x - 8y - 6 = 0

Hence the required equation of the line is 15x - 8y - 6 = 0.

Question 2 :

Find the x and y intercepts of the straight line

(i) 5x + 3y - 15 = 0 (ii) 2x - y + 16 = 0 (iii) 3x + 10y + 4 = 0

Solution :

(i) 5x + 3y - 15 = 0

To find the x and y intercepts from the given equation, we have to compare the given equation with the intercept form. 

Intercept form of a line :

(x/a) + (y/b) = 1

5x + 3y - 15 = 0

5x + 3 y = 15

Divide the equation by 15 

(5x/15) + (3y/15) = 15/15

(x/3) + (y/5) = 1

x-intercept = 3 and y -intercept = 5

(ii) 2x - y + 16 = 0 

(x/a) + (y/b) = 1

2x - y = -16

Divide the equation by -16 

(2x/(-16)) - (y/(-16)) = -16/(-16)

(x/(-8)) + (y/16) = 1

x-intercept = -8 and y -intercept = 16

(iii) 3x + 10y + 4 = 0

(x/a) + (y/b) = 1

3x + 10y = -4

Divide the equation by -4 

(3x/(-4)) + (10y/(-4)) = -4/(-4)

x/(-4/3) + y/(-4/10) = 1

x/(-4/3) + y/(-2/5) = 1

x-intercept = -4/3 and y -intercept = -2/5

Question 3 :

If graphed, the ordered pairs in the table above would form a line. Where would this line intersect the x-axis?

Solution :

By observing the table, one of the points on the line is (-1, 7) and y-intercept (when x = 0) is 5.

y = mx + b

x = -1, y = 7 and y-intercept (b) = 5

7 = m(-1) + 5

7 - 5 = -m

-m = 2

m = -2

y = -2x + 5

Question 4 :

Johanna picked 3 pounds of strawberries at a "pick-your-own" patch. At this particular patch, the cost is $1.50 for the pail and $3.99 per pound of strawberries picked. If a linear equation is created to represent the situation and written in the form y = mx+ b, which piece of the equation would the value 13.47 in this scenario most likely represent? 

a)  b      b)  m     c)  x     d)  y

Solution :

Cost of strawberries per pound = $3.99

Cost of 3 pounds of strawberries = 3(3.99)

= 11.97

Cost of pail = $1.50

Total cost = 11.97 + 1.50

y = 13.47

13.47 represents the value of variable y.

So, option d is correct.

Question 5 :

If the x-intercept of a line is positive and y-intercept is negative, does the line slant upward or downward from left to right? Explain your reasoning 

Solution :

x-intercept = positive

y-intercept = negative

The required line is slant upward.

Question 6 :

A student says that the x-intercept of the graph x + 2y = 5 is the point (0, 5), why is the student incorrect ?

Solution :

x + 2y = 5

To find the x-intercept, we have to apply y = 0

Applying y = 0, we get

x + 2(0) = 5

x = 5

So, the x-intercept is (5, 0). The given point is (0, 5), then the student is incorrect.

Question 7 :

At which point does the graph of the equation 2x + y = 4 cross the x-axis ?

Solution :

2x + y = 4

Finding the point where it cross the x-axis, then put y = 0

2x + 0 = 4

2x = 4

x = 4/2

x = 2

Question 8 :

What is the y-intercept of the graph of the equation 3x + y = 6 ?

Solution :

3x + y = 6 

To find y-intercept, put x = 0

3(0) + y = 6

0 + y = 6

y = 6

Question 9 :

Write the equation of a line in slope intercept form given a graph.

Solution :

Equation of line will be in the form y = mx + b

y-intercept (b) = -5

One of the point lies on the line is (3, -1)

-1 = m(3) - 5

-1 + 5 = 3m

3m = 4

m = 4/3

y = (4/3)x - 5

Question 10 :

Write the equation of a line in slope intercept form given a graph.

Solution :

Equation of line will be in the form y = mx + b

y-intercept (b) = 1

One of the point lies on the line is (4, -1)

-1 = m(4) + 1

-1 - 1 = 4m

4m = -2

m = -2/4

m = -1/2

y = (-1/2)x + 1

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