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

$$17x + my + 102 = 0$$
and
$$23x + 299y + 138 = 0$$
have an infinite number of solutions?

Question

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

$$17x + my + 102 = 0$$
and
$$23x + 299y + 138 = 0$$
have an infinite number of solutions?

Accepted Answer

**Solution:**

1. The given system of equations is:
   $$17x + my + 102 = 0$$
   $$23x + 299y + 138 = 0$$

2. For the system to have infinitely many solutions, the ratios of the coefficients must be equal:
   $$\frac{A}{P} = \frac{B}{Q} = \frac{C}{R}$$
   where the equations are in the form:
   $$Ax + By + C = 0$$ and $$Px + Qy + R = 0$$

3. Substitute the values:
   $$\frac{17}{23} = \frac{m}{299} = \frac{102}{138}$$

4. Calculate the ratio $$\frac{102}{138}$$:
   $$\frac{102}{138} = \frac{51}{69} = \frac{17}{23}$$

5. Since $$\frac{17}{23} = \frac{102}{138}$$, we only need to find *m* such that:
   $$\frac{m}{299} = \frac{17}{23}$$

6. Solve for *m*:
   $$m = \frac{17 	imes 299}{23}$$

7. Calculate:
   $$m = \frac{5083}{23} = 221$$

**Answer:** $221$

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