For what value of $m$ will the system of equations

$$18x - 72y + 13 = 0$$
and
$$7x - my - 17 = 0$$
have no solution?

Question

For what value of $m$ will the system of equations

$$18x - 72y + 13 = 0$$
and
$$7x - my - 17 = 0$$
have no solution?

Accepted Answer

**Solution:**

1. Write the equations in standard form:
   - Equation 1: $18x - 72y + 13 = 0$
   - Equation 2: $7x - my - 17 = 0$

2. For two linear equations $A_1x + B_1y + C_1 = 0$ and $A_2x + B_2y + C_2 = 0$, the condition for **no solution** (i.e., lines are parallel but not coincident) is:
   $$\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}$$

3. Identify coefficients:
   - $A_1 = 18, B_1 = -72, C_1 = 13$
   - $A_2 = 7, B_2 = -m, C_2 = -17$

4. Apply the condition for no solution:
   $$\frac{18}{7} = \frac{-72}{-m} \neq \frac{13}{-17}$$

5. Simplify the equality part:
   $$\frac{18}{7} = \frac{72}{m}$$

6. Cross-multiply to solve for $m$:
   $$18m = 7 \times 72$$
   $$18m = 504$$
   $$m = \frac{504}{18} = 28$$

7. Check the inequality part:
   - $\frac{18}{7} \approx 2.57$
   - $\frac{13}{-17} \approx -0.76$

Since $2.57 \neq -0.76$, the condition for no solution is satisfied.

**Answer:** $m = 28$

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