$ax + y = b$
$16x + ay = 24$
Suppose the values of a and b are chosen such that the system of linear equations produce multiple solutions. Then the product of a and b is __________. (answer in integer)
Understanding Multiple Solutions in Linear Systems
A system of two linear equations with two variables has multiple solutions (or infinite solutions) if the equations represent the same line. This occurs when the ratio of the coefficients of the corresponding variables and the constant terms are equal.
The given system is:
- $ax + y = b$
- $16x + ay = 24$
For multiple solutions, the following condition must hold:
$ \frac{\text{Coefficient of } x \text{ in Eq 1}}{\text{Coefficient of } x \text{ in Eq 2}} = \frac{\text{Coefficient of } y \text{ in Eq 1}}{\text{Coefficient of } y \text{ in Eq 2}} = \frac{\text{Constant term in Eq 1}}{\text{Constant term in Eq 2}} $Substituting the coefficients from the given equations:
$ \frac{a}{16} = \frac{1}{a} = \frac{b}{24} $Calculating Coefficient 'a'
From the first equality, we can find the possible values for 'a':
$ \frac{a}{16} = \frac{1}{a} $Cross-multiplying gives:
$ a \times a = 16 \times 1 $ $ a^2 = 16 $Solving for 'a':
$ a = \pm \sqrt{16} $ $ a = 4 \text{ or } a = -4 $Calculating Coefficient 'b' and Product 'ab'
Now, we use the equality involving 'a' and 'b' to find the relationship between them:
$ \frac{1}{a} = \frac{b}{24} $Cross-multiplying gives:
$ 1 \times 24 = a \times b $ $ ab = 24 $This equation directly gives the product of 'a' and 'b'. We can verify this for both values of 'a':
- If $a = 4$, then $4 \times b = 24$, so $b = 6$. The product $ab = 4 \times 6 = 24$.
- If $a = -4$, then $-4 \times b = 24$, so $b = -6$. The product $ab = (-4) \times (-6) = 24$.
In both cases, the product of 'a' and 'b' is 24.
Important Questions from System of Linear Equations
A system of equations is said to be inconsistent if
If a system of simultaneous equations has infinite solutions, then that system of equations is called:
Consider the system of simultaneous equation,
x + 2y + z = 6
2x + y + 2z = 6
x + y + z = 5
The system has,
The system of equations x + 2y = 13 and 3x + 6y = 9 has:
For what value of k, the system linear equation has no solution
(3k + 1)x + 3y - 2 = 0
(k2 + 1)x + (k - 2)y - 5 = 0
