If 4(x−y)=64 and 4(x+y)=1024, then find the value of x.
A1
B2
C3
D4
Answer:
D. 4
Read Explanation:
Convert Numbers to a Common Base
The given equations are $4^{(x - y)} = 64$ and $4^{(x + y)} = 1024$.
The first crucial step is to express the numbers on the right-hand side (64 and 1024) as powers of the base 4.
We know that $64 = 4 \times 4 \times 4 = 4^3$.
Similarly, $1024 = 4 \times 4 \times 4 \times 4 \times 4 = 4^5$.
Equate the Exponents
Now, substitute these values back into the original equations:
Equation 1 becomes: $4^{(x - y)} = 4^3$
Equation 2 becomes: $4^{(x + y)} = 4^5$
According to the law of exponents, if $a^m = a^n$ (where $a \ne 0, 1, -1$), then $m = n$.
Applying this rule, we can equate the exponents:
New Equation 3: $x - y = 3$
New Equation 4: $x + y = 5$
Solve the System of Linear Equations
We now have a simple system of two linear equations with two variables (x and y).
To find 'x', the easiest method is elimination. Add Equation 3 and Equation 4:
$(x - y) + (x + y) = 3 + 5$
$x - y + x + y = 8$
$2x = 8$
Finally, solve for 'x': $x = \frac{8}{2} = 4$.