1. Statement of the problem. Solve the system of linear equations: $$9D + 7E = 132$$ $$18D - 14E = 77$$ 2. Method and rules. We use the elimination method to remove one variable by combining equations. Important rules: align coefficients so one variable cancels when you add or subtract the equations. When dividing both sides or simplifying fractions we will show an intermediate line with \cancel{...} to mark canceled factors. 3. Multiply the first equation by 2 to match the $D$ coefficient of the second equation. $$2(9D + 7E) = 2(132)$$ $$18D + 14E = 264$$ 4. Write the two equations and add them to eliminate $E$. $$18D + 14E = 264$$ $$18D - 14E = 77$$ Adding the two equations gives: $$36D = 341$$ 5. Divide both sides by 36 to solve for $D$ and show cancellation. $$\frac{36D}{36} = \frac{341}{36}$$ $$\frac{\cancel{36}D}{\cancel{36}} = \frac{341}{36}$$ $$D = \frac{341}{36}$$ 6. Substitute $D$ into the first original equation to find $E$. $$9\left(\frac{341}{36}\right) + 7E = 132$$ Compute $9\cdot\frac{341}{36}$: $$\frac{3069}{36} + 7E = 132$$ Simplify the fraction $\frac{3069}{36}$ by canceling common factors step by step: $$\frac{3069}{36} = \frac{\cancel{3}1023}{\cancel{3}12}$$ $$\frac{1023}{12} = \frac{\cancel{3}341}{\cancel{3}4}$$ $$\frac{3069}{36} = \frac{341}{4}$$ So the substituted equation becomes: $$\frac{341}{4} + 7E = 132$$ Convert 132 to fourths and subtract $\frac{341}{4}$ from both sides: $$132 = \frac{528}{4}$$ $$7E = \frac{528}{4} - \frac{341}{4} = \frac{187}{4}$$ 7. Divide both sides by 7 to solve for $E$, showing cancellation. $$\frac{7E}{7} = \frac{187}{4 \cdot 7}$$ $$\frac{\cancel{7}E}{\cancel{7}} = \frac{187}{28}$$ $$E = \frac{187}{28}$$ 8. Final answer. $$D = \frac{341}{36}$$ $$E = \frac{187}{28}$$