for which values of a and b does the following pair of linear equation have an infinite number of solution ? 2x+3y=7 (a-b)x+(a+b)y=3a+b-2
To determine the values of "a" and "b" that make the given pair of linear equations have an infinite number of solutions, we need to find out when the two equations are dependent or parallel.
Let's start by rewriting the pair of equations in standard form:
Equation 1: 2x + 3y = 7 ----> 2x + 3y - 7 = 0
Equation 2: (a-b)x + (a+b)y = 3a + b - 2 ----> (a-b)x + (a+b)y - (3a + b - 2) = 0
For these equations to have an infinite number of solutions, they must represent the same line or be proportional. This means that the left side of Equation 2 should be a scalar multiple of Equation 1.
Let's equate the coefficients of x, y, and the constant term separately to determine the values of "a" and "b":
Comparing coefficients of x:
a - b = 2
Comparing coefficients of y:
a + b = 3
Comparing constant terms:
-3a - b + 2 = -7 (simplifying the right side of Equation 1)
Now, we can solve these three equations simultaneously to find the values of "a" and "b":
From the equation a - b = 2:
a = 2 + b -(Equation 4)
Plugging Equation 4 into the equation a + b = 3, we get:
(2 + b) + b = 3
2 + 2b = 3
2b = 3 - 2
2b = 1
b = 1/2 -(Equation 5)
Substituting the value of "b" back into Equation 4:
a = 2 + (1/2)
a = 5/2 -(Equation 6)
Therefore, the values of "a" and "b" that make the pair of linear equations have an infinite number of solutions are:
a = 5/2
b = 1/2
a-b=2
a+b=3
so
a=5/2
b=1/2
that gives
2x+3y=6
WAK! No solutions, rather than an infinite number. Is there a typo somewhere?