Analyzing the Algebraic Equations
We are given two equations:
- Equation 1: $X + Y = 2Z$
- Equation 2: $Z + W = 2X$
Our goal is to find a relationship between $X, Y, Z,$ and $W$ that must be true based on these equations.
Deriving Relationships Between Variables
First, let's isolate $Y$ from Equation 1:
$Y = 2Z - X$
Next, let's isolate $W$ from Equation 2:
$W = 2X - Z$
Testing the Relationship $X + Z = Y + W$
Now, let's see if the relationship presented in Option 1, $X + Z = Y + W$, is consistent with our derived expressions for $Y$ and $W$.
Consider the right side of Option 1: $Y + W$.
Substitute the expressions we found:
$Y + W = (2Z - X) + (2X - Z)$
Combine the terms:
- Combine the $Z$ terms: $2Z - Z = Z$
- Combine the $X$ terms: $-X + 2X = X$
So, $Y + W = Z + X$.
This means $X + Z = Y + W$.
Conclusion
The relationship $X + Z = Y + W$ is a direct consequence of the initial equations. This matches Option 1.
Similar Questions
- Given, $x + \frac{1}{x} = 5$, then determine the value of $\frac{2x}{x^2 - 1}$
If \(x + \frac{1}{x} = 2\sqrt{3}\), then find the value of \(x^{3} + \frac{1}{x^{3}}\).
Important Questions from Algebra
If 2x – y = 2 and xy = \(\frac{3}{2}\) , then what is the value of x 3– \(\frac{{{y^3}}}{8}\) ?
If (10a 3+ 4b 3) : (11a 3- 15b 3) = 7 : 5, then (3a + 5b) : (9a - 2b) =?
- If 4sin 2 θ = 3(1+ cos θ), 0° < θ < 90°, then what is the value of (2tan θ + 4sin θ - sec θ)?
The value of:
\(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\)
If (x + y) 3+ 27(x - y) 3= (Ax - 2y)(Bx 2+ Cxy + 13y 2), then the value of A - B - C is:
