1. **State the problem:** We are given the system of equations: $$2x - ky = 14$$ $$5x - 2y = 5$$ We need to find the values of the constant $k$ for which this system has no solution. 2. **Recall the condition for no solution:** A system of two linear equations has no solution if the lines are parallel but not coincident. This means their slopes are equal but their intercepts differ. 3. **Rewrite each equation in slope-intercept form $y = mx + b$:** - From the first equation: $$2x - ky = 14 \implies -ky = 14 - 2x \implies y = \frac{2}{k}x - \frac{14}{k}$$ - From the second equation: $$5x - 2y = 5 \implies -2y = 5 - 5x \implies y = \frac{5}{2}x - \frac{5}{2}$$ 4. **Set the slopes equal for parallel lines:** $$\frac{2}{k} = \frac{5}{2}$$ 5. **Solve for $k$:** $$2 \times 2 = 5 \times k \implies 4 = 5k \implies k = \frac{4}{5}$$ 6. **Check intercepts to ensure no solution:** - Intercept of first line: $-\frac{14}{k} = -\frac{14}{4/5} = -\frac{14 \times 5}{4} = -\frac{70}{4} = -17.5$ - Intercept of second line: $-\frac{5}{2} = -2.5$ Since the intercepts are different, the lines are parallel and distinct, so the system has no solution when: $$k = \frac{4}{5}$$