Consider the system of equations: x + y = 2 and 2x + 2y = 5. This system has
No solution
Analyzing the System of Equations
We are given a system of two linear equations:
- \(x + y = 2\)
- \(2x + 2y = 5\)
We need to determine how many solutions this system has.
Comparing the Equations
Let's look closely at the two equations. We can compare their coefficients.
In the first equation, the coefficients are 1 for x, 1 for y, and the constant is 2.
In the second equation, the coefficients are 2 for x, 2 for y, and the constant is 5.
Notice that the coefficients of x and y in the second equation are exactly twice the coefficients of x and y in the first equation:
- Coefficient of x: \(2 = 2 \times 1\)
- Coefficient of y: \(2 = 2 \times 1\)
Let's see what happens if we multiply the first equation by 2:
\(2 \times (x + y) = 2 \times 2\)
\(2x + 2y = 4\)
Identifying Inconsistency
Now we have the modified first equation \(2x + 2y = 4\) and the original second equation \(2x + 2y = 5\).
Both equations have the same expression on the left side (\(2x + 2y\)), but they are set equal to different constants on the right side (4 and 5).
This creates a contradiction:
\(2x + 2y = 4\)
\(2x + 2y = 5\)
This implies that \(4 = 5\), which is impossible.
Since there is a contradiction, there are no values of x and y that can satisfy both equations simultaneously.
Geometric Interpretation
Geometrically, each linear equation represents a straight line in a two-dimensional coordinate system.
When we have a system of two linear equations, the solution(s) correspond to the point(s) where the lines intersect.
For the given equations:
- From \(x + y = 2\), we can write \(y = -x + 2\). The slope is \(-1\) and the y-intercept is 2.
- From \(2x + 2y = 5\), we can write \(2y = -2x + 5\), which simplifies to \(y = -x + \frac{5}{2}\). The slope is \(-1\) and the y-intercept is \(\frac{5}{2}\).
Since both lines have the same slope (\(-1\)) but different y-intercepts (2 and \(\frac{5}{2}\)), the lines are parallel and distinct.
Parallel lines never intersect.
Conclusion on the Number of Solutions
Because the lines represented by the equations are parallel and do not intersect, the system of equations has no common solution.
A system of linear equations with no solution is called an inconsistent system.
Important Questions from Matrix Algebra
The standard ordered basis of R 3 is {e 1, e 2, e 3} Let T : R 3 → R 3 be the linear transformation such that T(e 1) = 7e 1- 5e 3, T (e 2) = -2e 2+ 9e 3, T(e 3) = e 1+ e 2+ e 3. The standard matrix of T is:
The system of equations
x + y + z = 6;
x + 4y + 6z = 20;
x + 4y + λz = μ
has NO solution for values of λ and μ given by
What is the transformation matrix M that transforms a square in the xy-plane defined by (1, 1) T, (-1, 1) T, (-1, -1) T and (1, -1) T to a parallelogram whose corresponding vertices are (2, 1) T, (0, 1) T, (-2, -1) T and (0, -1) T?
- If A = \( \left[\begin{array}{cc}0 & 1 \\ −1 & 0\end{array}\right]\) and (aI 2 + bA)2 = A, then
The rank of the matrix \(\begin{bmatrix} 1 & 1 & 1 \\\ a & b & c \\\ a^2 & b^2 & c^2 \end{bmatrix}\) where a = b ≠ c is:
