Math Problem Statement
how did they get those values of k and h
Solution
To determine the values of and for which the given system of linear equations has infinitely many solutions, we need to check the conditions under which the system is dependent.
The system of equations is:
4x - 5y - 5z &= 4 \\ 7x + 9y - 3z &= 5 \\ -18x - 13y + hz &= k \end{aligned}$$ For a system to have infinitely many solutions, the third equation must be a linear combination of the first two equations. This implies that the coefficients of $$x$$, $$y$$, and $$z$$ in the third equation must be proportional to the coefficients in the first two equations. ### Step-by-step process: 1. **Set up proportionality condition:** The third equation should be a combination of the first two. We write this as: $$(-18x - 13y + hz) = \alpha (4x - 5y - 5z) + \beta (7x + 9y - 3z)$$ for some constants $$\alpha$$ and $$\beta$$. 2. **Match coefficients:** - For the $$x$$-terms: $$-18 = 4\alpha + 7\beta$$ - For the $$y$$-terms: $$-13 = -5\alpha + 9\beta$$ - For the $$z$$-terms: $$h = -5\alpha - 3\beta$$ - For the constant terms (right-hand side): $$k = 4\alpha + 5\beta$$ 3. **Solve the system of equations for $$\alpha$$ and $$\beta$$:** Using the first two equations: - From the $$x$$-term equation: $$4\alpha + 7\beta = -18$$ - From the $$y$$-term equation: $$-5\alpha + 9\beta = -13$$ Solve these two simultaneous equations to find $$\alpha$$ and $$\beta$$. 4. **Calculate $$h$$ and $$k$$:** Once you have $$\alpha$$ and $$\beta$$, substitute them into the equations for $$h$$ and $$k$$ to determine their values. ### Solution: After solving this system of equations, you should find that $$k = -14$$ and $$h = 11$$, which are the values that make the system have infinitely many solutions. Would you like more details on how to solve the system step by step? Here are 5 related questions you might be interested in: 1. How do you check if a system of linear equations is consistent or inconsistent? 2. What are the conditions for a system of equations to have a unique solution? 3. How does the rank of a matrix relate to the number of solutions in a system of linear equations? 4. Can you use matrix methods (like Gaussian elimination) to find $$k$$ and $$h$$ in this case? 5. What is the geometric interpretation of having infinitely many solutions? **Tip:** A system of linear equations has infinitely many solutions when the equations represent the same plane or are dependent. This usually occurs when one equation is a linear combination of the others.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems of Equations
Proportionality
Linear Combinations
Formulas
4x - 5y - 5z = 4
7x + 9y - 3z = 5
-18x - 13y + hz = k
Proportionality condition: (-18x - 13y + hz) = α(4x - 5y - 5z) + β(7x + 9y - 3z)
Theorems
Condition for infinite solutions: One equation is a linear combination of others
Suitable Grade Level
Grades 10-12
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