To solve this problem, let's define variables to represent the unknowns:

Let:

• $x$ = the number of nails used to make a small bookshelf
• $y$ = the number of nails used to make a large bookshelf

Step 1: Translate the situation into a system of equations.

From the first piece of information:

• Nate built 2 small bookshelves and 4 large bookshelves, using a total of 280 nails. This gives us the equation: $2x + 4y = 280$

From the second piece of information:

• Nate built 10 small bookshelves and 4 large bookshelves, using a total of 568 nails. This gives us the equation: $10x + 4y = 568$

So, the system of equations is: 2x + 4y = 280 \tag{1} 10x + 4y = 568 \tag{2}

Step 2: Solve using an augmented matrix.

We can represent this system as an augmented matrix:

2 & 4 & | & 280 \\ 10 & 4 & | & 568 \end{bmatrix}$$ Now, let's solve it using row operations. 1. **Row 1:** $$[2, 4 | 280]$$ and **Row 2:** $$[10, 4 | 568]$$ 2. Subtract $$5 \times$$ Row 1 from Row 2 to eliminate $$y$$: $$R_2 \to R_2 - 5R_1$$ This gives us: $$\begin{bmatrix} 2 & 4 & | & 280 \\ 0 & -16 & | & -1320 \end{bmatrix}$$ 3. Now divide Row 2 by $$-16$$ to simplify: $$R_2 \to \frac{1}{-16} R_2$$ Resulting in: $$\begin{bmatrix} 2 & 4 & | & 280 \\ 0 & 1 & | & 82.5 \end{bmatrix}$$ 4. Substitute $$y = 82.5$$ back into the first equation to solve for $$x$$: $$2x + 4(82.5) = 280$$ $$2x + 330 = 280$$ $$2x = 280 - 330 = -50$$ $$x = \frac{-50}{2} = -25$$ ### Step 3: Interpret the results. Thus, the solution to the system is: - $$x = -25$$, which means Nate uses 25 nails to make each small bookshelf. - $$y = 82.5$$, which means Nate uses 82.5 nails to make each large bookshelf. ### Conclusion: Nate uses 25 nails to make each small bookshelf and 82.5 nails to make each large bookshelf. ### Do you want a deeper explanation of any of these steps, or have any other questions? Here are five related questions: 1. How do you interpret the negative value for $$x$$ in this context? 2. What would happen if the coefficients in the equations were different? 3. Can this method be applied to any system of linear equations? 4. How would you solve this system using substitution instead of an augmented matrix? 5. What is the significance of using an augmented matrix in solving systems of equations? **Tip:** Always check your solution by substituting it back into the original system of equations to ensure it satisfies both equations.