Which of the following pairs of linear equations are consistent or inconsistent? If consistent, obtain the solution graphically.
(i) \(x + y = 5\); \(2x + 2y = 10\)
(ii) \(x - y = 8\); \(3x - 3y = 16\)
(iii) \(2x + y - 6 = 0\); \(4x - 2y - 4 = 0\)
(iv) \(2x - 2y - 2 = 0\); \(4x - 4y - 5 = 0\)
Use the criteria based on \( \dfrac{a_1}{a_2}, \dfrac{b_1}{b_2}, \dfrac{c_1}{c_2} \).
(i) \(x + y = 5\) and \(2x + 2y = 10\)
Second equation simplifies by dividing through by 2:
\[ 2x + 2y = 10 \Rightarrow x + y = 5. \]
Thus both equations represent the same line. Here
\[ \dfrac{a_1}{a_2} = \dfrac{1}{2}, \quad \dfrac{b_1}{b_2} = \dfrac{1}{2}, \quad \dfrac{c_1}{c_2} = \dfrac{-5}{-10} = \dfrac{1}{2}. \]
All ratios are equal, so the pair is consistent with infinitely many solutions.
Graphical solution: Both equations give the same line \(x + y = 5\). Every point on this line, such as \((0,5), (5,0), (2,3)\), is a solution.
(ii) \(x - y = 8\) and \(3x - 3y = 16\)
Write in standard form: \(x - y - 8 = 0\); \(3x - 3y - 16 = 0\).
Ratios:
\[ \dfrac{a_1}{a_2} = \dfrac{1}{3}, \quad \dfrac{b_1}{b_2} = \dfrac{-1}{-3} = \dfrac{1}{3}, \quad \dfrac{c_1}{c_2} = \dfrac{-8}{-16} = \dfrac{1}{2}. \]
We have \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\); therefore the lines are parallel and distinct.
The pair is inconsistent (no solution).
(iii) \(2x + y - 6 = 0\) and \(4x - 2y - 4 = 0\)
Rewrite the second equation:
\[ 4x - 2y - 4 = 0. \]
Here \(a_1 = 2, b_1 = 1, c_1 = -6;\) \(a_2 = 4, b_2 = -2, c_2 = -4.\)
Ratios:
\[ \dfrac{a_1}{a_2} = \dfrac{2}{4} = \dfrac{1}{2}, \quad \dfrac{b_1}{b_2} = \dfrac{1}{-2} = -\dfrac{1}{2}. \]
Since \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\), the lines intersect at a unique point, so the pair is consistent.
Solving algebraically for the point of intersection (which will match the graphical solution):
From \(2x + y - 6 = 0\), \(y = 6 - 2x\).
Substitute into \(4x - 2y - 4 = 0\):
\[ 4x - 2(6 - 2x) - 4 = 0 \Rightarrow 4x - 12 + 4x - 4 = 0 \Rightarrow 8x - 16 = 0 \Rightarrow x = 2. \]
Then \(y = 6 - 2x = 6 - 4 = 2\).
Graphically, the lines intersect at \((2,2)\).
(iv) \(2x - 2y - 2 = 0\) and \(4x - 4y - 5 = 0\)
Here \(a_1 = 2, b_1 = -2, c_1 = -2;\) \(a_2 = 4, b_2 = -4, c_2 = -5.\)
Ratios:
\[ \dfrac{a_1}{a_2} = \dfrac{2}{4} = \dfrac{1}{2}, \quad \dfrac{b_1}{b_2} = \dfrac{-2}{-4} = \dfrac{1}{2}, \quad \dfrac{c_1}{c_2} = \dfrac{-2}{-5} = \dfrac{2}{5}. \]
Since \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\), the lines are parallel and distinct.
The pair is inconsistent (no solution).
Use the ratio test to label each pair as intersecting, parallel, or coincident. For intersecting pairs, either plot the lines or solve one equation for a variable and substitute to find the intersection point.