1. **Problem Statement:** A man buys 20 pens and 30 pencils for a total of 600. He also buys 30 pens and 10 pencils for a total of 550. We need to find the price of one pen (x) and one pencil (y). 2. **Forming the equations:** Let price of one pen = $x$ Rs and one pencil = $y$ Rs. From the problem: - $20x + 30y = 600$ - $30x + 10y = 550$ These are linear equations in standard form $Ax + By = C$. 3. **Graphical Method:** - Rewrite each equation to express $y$ in terms of $x$: - From $20x + 30y = 600$, $$30y = 600 - 20x \Rightarrow y = \frac{600 - 20x}{30} = 20 - \frac{2}{3}x$$ - From $30x + 10y = 550$, $$10y = 550 - 30x \Rightarrow y = \frac{550 - 30x}{10} = 55 - 3x$$ - Plot these two lines on a graph with $x$ (pens) on the horizontal axis and $y$ (pencils) on the vertical axis. - The point where the two lines intersect gives the solution $(x,y)$. 4. **Substitution Method:** - From the first equation, express $y$: $$y = 20 - \frac{2}{3}x$$ - Substitute into the second equation: $$30x + 10\left(20 - \frac{2}{3}x\right) = 550$$ - Simplify: $$30x + 200 - \frac{20}{3}x = 550$$ - Multiply entire equation by 3 to clear denominator: $$90x + 600 - 20x = 1650$$ - Combine like terms: $$70x + 600 = 1650$$ - Subtract 600: $$70x = 1050$$ - Divide by 70: $$x = 15$$ - Substitute $x=15$ back into $y = 20 - \frac{2}{3}x$: $$y = 20 - \frac{2}{3} \times 15 = 20 - 10 = 10$$ 5. **Elimination Method:** - Equations: $$20x + 30y = 600$$ $$30x + 10y = 550$$ - Multiply first equation by 1 and second by 3 to align $y$ coefficients: $$20x + 30y = 600$$ $$90x + 30y = 1650$$ - Subtract first from second: $$(90x - 20x) + (30y - 30y) = 1650 - 600$$ $$70x = 1050$$ - Solve for $x$: $$x = 15$$ - Substitute $x=15$ into first equation: $$20(15) + 30y = 600$$ $$300 + 30y = 600$$ $$30y = 300$$ $$y = 10$$ 6. **Total cost for 50 pens and 40 pencils:** - Using $x=15$ and $y=10$: $$\text{Total cost} = 50 \times 15 + 40 \times 10 = 750 + 400 = 1150$$ **Final answer:** - Price of one pen = Rs. 15 - Price of one pencil = Rs. 10 - Total cost for 50 pens and 40 pencils = Rs. 1150 All steps are shown clearly with formulas and calculations.