4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) , (ii) , (iii) , (iv) ,

Understand the Question To check if a pair of linear equations and is consistent (has at least one solution) or inconsistent (has no solution), compare the ratios of their coefficients: Intersecting Lines: Consistent (unique solution) Coincident Lines: Consistent (infinitely many solutions) Parallel Lines: Inconsistent (no solution) If the system is consistent, we find the solution graphically by plotting coordinate points for each line. Part (i) , Step 1 · Compare Ratios to Check Consistency Writing equations in standard form : Comparing the ratios of the coefficients: Since , the lines are coincident and the pair is consistent with infinitely many solutions. Step 2 · Plot the Lines Graphically For : If , If , The equation simplifies to , so both equations represent the same line. All points on this line are solutions. Answer > (i) Consistent (infinitely many solutions, coincident lines) Part (ii) , Step 1 · Compare Ratios to Check Consistency Writing equations in standard form: Comparing the ratios: Since , the lines are parallel. Therefore, the pair is inconsistent (no solution). Answer > (ii) Inconsistent (no solution) Part (iii) , Step 1 · Compare Ratios to Check Consistency Given equations: Comparing the ratios: Since , the pair of equations is consistent with a unique solution. Step 2 · Find Coordinates and Solve Graphically For : If , If , For : If , If , Plotting these points on a graph, the two lines intersect at . Answer > (iii) Consistent; the graphical solution is Part (iv) , Step 1 · Compare Ratios to Check Consistency Given equations: Comparing the ratios: Since , the lines are parallel. Therefore, the pair is inconsistent (no solution). Answer > (iv) Inconsistent (no solution) Common Mistakes Sign Errors in Standard Form: Ensure constant terms and are on the same side of the equation (either both on LHS or both on RHS) before taking ratios. Sign in Ratios: Pay close attention to negative signs; for example, , which indicates intersecting lines (consistent) rather than parallel lines.

Pair of Linear Equations in Two Variables | Exercise 3.1

Question 4

Which of the following pairs of linear equations are consistent/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) $2x - 2y - 2 = 0$ , $4x - 4y - 5 = 0$

Solution

Understand the Question

To check if a pair of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is consistent (has at least one solution) or inconsistent (has no solution), compare the ratios of their coefficients:

Intersecting Lines: $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2} \implies$ Consistent (unique solution)
Coincident Lines: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \implies$ Consistent (infinitely many solutions)
Parallel Lines: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \implies$ Inconsistent (no solution)

If the system is consistent, we find the solution graphically by plotting coordinate points for each line.

(i) $x + y = 5$ , $2x + 2y = 10$

Step 1 · Compare Ratios to Check Consistency

Writing equations in standard form $ax + by + c = 0$ :

x + y - 5 = 0 \implies a_1 = 1, \, b_1 = 1, \, c_1 = -5

2x + 2y - 10 = 0 \implies a_2 = 2, \, b_2 = 2, \, c_2 = -10

Comparing the ratios of the coefficients:

\begin{aligned} \dfrac{a_1}{a_2} &= \dfrac{1}{2} \\[0.6em] \dfrac{b_1}{b_2} &= \dfrac{1}{2} \\[0.6em] \dfrac{c_1}{c_2} &= \dfrac{-5}{-10} = \dfrac{1}{2} \end{aligned}

Since $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$ , the lines are coincident and the pair is consistent with infinitely many solutions.

Step 2 · Plot the Lines Graphically

For $x + y = 5$ :

If $x = 0$ , $y = 5 \implies (0, 5)$
If $y = 0$ , $x = 5 \implies (5, 0)$

The equation $2x + 2y = 10$ simplifies to $x + y = 5$ , so both equations represent the same line. All points on this line are solutions.

Answer

(i) Consistent (infinitely many solutions, coincident lines)

(ii) $x - y = 8$ , $3x - 3y = 16$

Step 1 · Compare Ratios to Check Consistency

Writing equations in standard form:

x - y - 8 = 0 \implies a_1 = 1, \, b_1 = -1, \, c_1 = -8

3x - 3y - 16 = 0 \implies a_2 = 3, \, b_2 = -3, \, c_2 = -16

Comparing the ratios:

\begin{aligned} \dfrac{a_1}{a_2} &= \dfrac{1}{3} \\[0.6em] \dfrac{b_1}{b_2} &= \dfrac{-1}{-3} = \dfrac{1}{3} \\[0.6em] \dfrac{c_1}{c_2} &= \dfrac{-8}{-16} = \dfrac{1}{2} \end{aligned}

Since $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$ , the lines are parallel. Therefore, the pair is inconsistent (no solution).

Answer

(ii) Inconsistent (no solution)

(iii) $2x + y - 6 = 0$ , $4x - 2y - 4 = 0$

Step 1 · Compare Ratios to Check Consistency

Given equations:

2x + y - 6 = 0 \implies a_1 = 2, \, b_1 = 1, \, c_1 = -6

4x - 2y - 4 = 0 \implies a_2 = 4, \, b_2 = -2, \, c_2 = -4

Comparing the ratios:

\begin{aligned} \dfrac{a_1}{a_2} &= \dfrac{2}{4} = \dfrac{1}{2} \\[0.6em] \dfrac{b_1}{b_2} &= \dfrac{1}{-2} = -\dfrac{1}{2} \end{aligned}

Since $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$ , the pair of equations is consistent with a unique solution.

Step 2 · Find Coordinates and Solve Graphically

For $2x + y - 6 = 0 \implies y = 6 - 2x$ :

If $x = 0$ , $y = 6 \implies (0, 6)$
If $x = 3$ , $y = 0 \implies (3, 0)$

For $4x - 2y - 4 = 0 \implies y = 2x - 2$ :

If $x = 0$ , $y = -2 \implies (0, -2)$
If $x = 1$ , $y = 0 \implies (1, 0)$ Plotting these points on a graph, the two lines intersect at $(2, 2)$ .

Answer

(iii) Consistent; the graphical solution is $(2, 2)$

(iv) $2x - 2y - 2 = 0$ , $4x - 4y - 5 = 0$

Step 1 · Compare Ratios to Check Consistency

Given equations:

2x - 2y - 2 = 0 \implies a_1 = 2, \, b_1 = -2, \, c_1 = -2

4x - 4y - 5 = 0 \implies a_2 = 4, \, b_2 = -4, \, c_2 = -5

Comparing the ratios:

\begin{aligned} \dfrac{a_1}{a_2} &= \dfrac{2}{4} = \dfrac{1}{2} \\[0.6em] \dfrac{b_1}{b_2} &= \dfrac{-2}{-4} = \dfrac{1}{2} \\[0.6em] \dfrac{c_1}{c_2} &= \dfrac{-2}{-5} = \dfrac{2}{5} \end{aligned}

Since $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$ , the lines are parallel. Therefore, the pair is inconsistent (no solution).

Answer

(iv) Inconsistent (no solution)

Common Mistakes

Sign Errors in Standard Form: Ensure constant terms $c_1$ and $c_2$ are on the same side of the equation (either both on LHS or both on RHS) before taking ratios.
Sign in Ratios: Pay close attention to negative signs; for example, $\dfrac{1}{2} \neq -\dfrac{1}{2}$ , which indicates intersecting lines (consistent) rather than parallel lines.

Question 4

More questions in Exercise 3.1