For the system a_1x + b_1y + c_1 = 0 and a_2x + b_2y + c_2 = 0, we have a_1 = 2, b_1 = 1, c_1 = -6 and a_2 = 4, b_2 = -2, c_2 = -4. Since (a_1)/(a_2) ≠ (b_1)/(b_2), the lines intersect at exactly one point, meaning the pair of equations is consistent with a unique solution.

For the system a_1x + b_1y + c_1 = 0 and a_2x + b_2y + c_2 = 0, we have a_1 = 2, b_1 = 1, c_1 = -6 and a_2 = 4, b_2 = -2, c_2 = -4. Since (a_1)/(a_2) ≠ (b_1)/(b_2), the lines intersect at exactly one point, meaning the pair of equations is consistent with a unique solution.

4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, have solutions, find them 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) $2x - 2y - 2 = 0$ , $4x - 4y - 5 = 0$

Answer: The pair of linear equations is consistent, and the solution from the graph is $x = 2$ , $y = 2$ .

Step-by-step solution

Step 1: Compare the ratios of coefficients

For the system $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ , we have $a_1 = 2$ , $b_1 = 1$ , $c_1 = -6$ and $a_2 = 4$ , $b_2 = -2$ , $c_2 = -4$ . Since $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ , the lines intersect at exactly one point, meaning the pair of equations is consistent with a unique solution.

Step 2: Find coordinates for the first line

From the first equation $2x + y - 6 = 0$ , expressing $y$ gives $y = 6 - 2x$ . Substituting $x = 0$ gives $y = 6$ ; substituting $x = 2$ gives $y = 2$ ; and substituting $x = 3$ gives $y = 0$ .

Step 3: Find coordinates for the second line

From the second equation $4x - 2y - 4 = 0$ , dividing by $2$ yields $2x - y - 2 = 0$ , which gives $y = 2x - 2$ . Substituting $x = 0$ gives $y = -2$ ; substituting $x = 1$ gives $y = 0$ ; and substituting $x = 2$ gives $y = 2$ .

Step 4: Determine the solution graphically

Plotting both sets of points and drawing straight lines through them, we observe that the two lines intersect at the point $(2, 2)$ . Thus, the graphical solution to the pair of equations is $x = 2$ and $y = 2$ .

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