the system of linear equations below has a unique solution for all but one value of a :
5 x - 5 y = 25
48 x + a y = 240
What is this exceptional value for a ?
If the two lines would be parallel, we could not find a solution,
that happens when
5/48 = -5/a
5a = -5(48)
a = -48
Oh, linear equations, what a maze! Let me put my clown hat on and help you find the exceptional value for a.
First, let's take a look at the slope (coefficient) of y in the second equation: "a". The slope affects the uniqueness of the solution.
If we want a unique solution, the slopes should be different for the x and y terms. So, the exceptional value for a would be when the slope of "a y" in the second equation is equal to the slope of "5 x" in the first equation.
Therefore, to find this exceptional value, we set:
a = 5 / 48
Tada! The exceptional value for a is 5/48. I hope this answer brings a smile to your face!
To find the exceptional value for a in the system of linear equations, we need to determine when the system has a unique solution.
Let's solve the system of equations using the method of substitution.
Step 1: Solve the first equation for x in terms of y.
5x - 5y = 25
5x = 5y + 25
x = y + 5
Step 2: Substitute the expression for x into the second equation.
48x + ay = 240
48(y + 5) + ay = 240
48y + 240 + ay = 240
48y + ay = 0
Step 3: Combine like terms.
(48 + a) y = 0
Step 4: For the system to have a unique solution, the coefficient of y must be non-zero. So, set (48 + a) ≠ 0 and solve for a.
48 + a ≠ 0
a ≠ -48
Therefore, the exceptional value for a is -48.
To find the exceptional value for a, we need to determine when the system of linear equations has no unique solution or no solution at all.
We can start by using the method of elimination to solve the equations simultaneously. Let's multiply the first equation by 48 and the second equation by 5 to eliminate the variables x:
(48)(5x - 5y) = (48)(25)
(5)(48x + ay) = (5)(240)
This simplifies the equations to:
240x - 240y = 1200
240x + 5ay = 1200
Next, subtract the second equation from the first to eliminate x:
(240x - 240y) - (240x + 5ay) = 1200 - 1200
-240y - 5ay = 0
-240(y + ay/5) = 0
Now, we can see that for the system to have no unique solution or no solution at all, the expression y + ay/5 must equal zero, or in other words:
y(1 + a/5) = 0
For this equation to be true, one of two possibilities must occur: either y = 0 or 1 + a/5 = 0.
1) If y = 0, then substituting this into the first equation gives:
5x = 25
x = 5
Thus, the solution to the system is x = 5 and y = 0.
2) If 1 + a/5 = 0, then solving for a gives:
a/5 = -1
a = -5
Therefore, the exceptional value for a is -5. If the value of a is -5, then the system of linear equations will have no unique solution or no solution at all. For all other values of a, the system will have a unique solution.