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The number of integral values of m for which the x coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer is
- a)2
- b)0
- c)4
- d)1
Correct answer is option 'A'. Can you explain this answer?
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Given:
The lines are represented by the equations:
1) 3x - 4y = 9
2) y = mx - 1
To find:
The number of integral values of m for which the x-coordinate of the point of intersection of the lines is also an integer.
Solution:
To find the x-coordinate of the point of intersection, we need to find the values of x and y that satisfy both equations simultaneously.
Step 1: Substitute y in equation 1:
Substituting the value of y from equation 2 into equation 1, we get:
3x - 4(mx - 1) = 9
Simplifying the equation, we have:
3x - 4mx + 4 = 9
-4mx + 3x = 5
x(3 - 4m) = 5 - 4
x(3 - 4m) = 1
Step 2: Find the possible values of m:
For the x-coordinate to be an integer, the denominator (3 - 4m) must divide the numerator (1) evenly. This means that (3 - 4m) must be a factor of 1.
The factors of 1 are ±1. So we have two cases:
Case 1: 3 - 4m = 1
Solving this equation for m, we get:
-4m = 1 - 3
-4m = -2
m = -2/-4
m = 1/2
Case 2: 3 - 4m = -1
Solving this equation for m, we get:
-4m = -1 - 3
-4m = -4
m = -4/-4
m = 1
Step 3: Check if the values of m are integral:
In case 1, m = 1/2, which is not an integer.
In case 2, m = 1, which is an integer.
Conclusion:
There is only 1 integral value of m for which the x-coordinate of the point of intersection is also an integer. Therefore, the correct answer is option A) 1.
The lines are represented by the equations:
1) 3x - 4y = 9
2) y = mx - 1
To find:
The number of integral values of m for which the x-coordinate of the point of intersection of the lines is also an integer.
Solution:
To find the x-coordinate of the point of intersection, we need to find the values of x and y that satisfy both equations simultaneously.
Step 1: Substitute y in equation 1:
Substituting the value of y from equation 2 into equation 1, we get:
3x - 4(mx - 1) = 9
Simplifying the equation, we have:
3x - 4mx + 4 = 9
-4mx + 3x = 5
x(3 - 4m) = 5 - 4
x(3 - 4m) = 1
Step 2: Find the possible values of m:
For the x-coordinate to be an integer, the denominator (3 - 4m) must divide the numerator (1) evenly. This means that (3 - 4m) must be a factor of 1.
The factors of 1 are ±1. So we have two cases:
Case 1: 3 - 4m = 1
Solving this equation for m, we get:
-4m = 1 - 3
-4m = -2
m = -2/-4
m = 1/2
Case 2: 3 - 4m = -1
Solving this equation for m, we get:
-4m = -1 - 3
-4m = -4
m = -4/-4
m = 1
Step 3: Check if the values of m are integral:
In case 1, m = 1/2, which is not an integer.
In case 2, m = 1, which is an integer.
Conclusion:
There is only 1 integral value of m for which the x-coordinate of the point of intersection is also an integer. Therefore, the correct answer is option A) 1.
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The number of integral values of m for which the x coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer is a)2b)0c)4d)1Correct answer is option 'A'. Can you explain this answer?
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