Math Problem Statement
Solution
Let's address both questions presented in the image.
Question 3: Coin Problem
The jar contains pennies, nickels, and dimes, with a total of 11 coins and a total value of $0.87. We need to find how many of each coin type there are.
Let:
- be the number of pennies,
- be the number of nickels,
- be the number of dimes.
We have the following two equations based on the problem:
-
The total number of coins:
-
The total value of the coins:
We can multiply the second equation by 100 to eliminate decimals:
Now we have a system of equations:
We can solve this system for , , and .
Question 4: Homogeneous System of Linear Equations
The given system of linear equations is:
To solve this, we first convert it into an augmented matrix:
2 & -6 & 4 & -8 & 0 \\ -1 & 4 & -4 & 9 & 0 \\ 3 & -7 & 3 & -5 & 0 \end{pmatrix}$$ We then apply row-reduction to this augmented matrix to find its row-echelon form. If it leads to an inconsistency (such as a row of all zeros in the coefficients but a nonzero number on the right), the system will have no solution. I'll solve both of these now. ### Solving the coin problem first: Let me solve the system of equations for Question 3.The solution to the coin problem shows that the number of pennies ($$p$$) and nickels ($$n$$) are dependent on the number of dimes ($$d$$). Specifically: $$p = \frac{5d}{4} - 8$$ $$n = 19 - \frac{9d}{4}$$ Since $$p$$, $$n$$, and $$d$$ must be non-negative integers, we need to test values for $$d$$ (the number of dimes) that make $$p$$ and $$n$$ both integers. Let's check possible integer solutions for $$d$$. Now, I'll move on to solve the system of linear equations in Question 4 by performing Gaussian elimination on the matrix.The row-echelon form of the augmented matrix for the system of linear equations is: $$\begin{pmatrix} 1 & 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & -3 & 0 \\ \end{pmatrix}$$ This represents the following system of equations: $$x_1 - x_4 = 0$$ $$x_2 - x_4 = 0$$ $$x_3 - 3x_4 = 0$$ From this, we can see that: - $$x_1 = x_4$$ - $$x_2 = x_4$$ - $$x_3 = 3x_4$$ Therefore, the solution to the system is dependent on the free variable $$x_4$$, meaning the system has infinitely many solutions. If you need further clarification or a detailed breakdown of any part, let me know! ### 5 Related Questions: 1. How can we handle cases where a system has infinitely many solutions? 2. What are the conditions for a system of linear equations to have no solution? 3. How can we identify if a matrix is singular using Gaussian elimination? 4. How do you interpret the relationship between the variables in homogeneous systems? 5. Can we solve for systems of higher dimensions in a similar manner? ### Tip: Always check the row-echelon form of a system's matrix to determine if the system has a unique solution, infinitely many solutions, or no solution at all.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Linear Equations
Formulas
p + n + d = 11
0.01p + 0.05n + 0.10d = 0.87
Theorems
Row-Echelon Form
Gaussian Elimination
Suitable Grade Level
Grades 8-10
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