Math Problem Statement
Is the ordered pair (-3, 5) a solution of the system of equations? Is the ordered pair (3/5, 3) a solution of the equation 7x + 9y = 37?
Solution
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Question 1:
The problem asks whether the given ordered pair is a solution of the given system of equations:
2 & -3 \\ 3 & 5 \end{matrix} \right] \left( \begin{matrix} x \\ y \end{matrix} \right) = \left( \begin{matrix} -3 \\ 8 \end{matrix} \right)$$ ### Solution Approach: Substitute $$x = -3$$ and $$y = 5$$ into the system: 1. For the first equation: $$2x - 3y = -3$$ $$2(-3) - 3(5) = -6 - 15 = -21 \quad (\text{not equal to } -3)$$ So, the first equation does not hold. Thus, $$\left( \begin{matrix} -3 \\ 5 \end{matrix} \right)$$ is **not** a solution. ### Question 2: Is the ordered pair $$\left( \frac{3}{5}, 3 \right)$$ a solution of the equation $$7x + 9y = 37$$? Substitute $$x = \frac{3}{5}$$ and $$y = 3$$ into the equation: $$7 \left(\frac{3}{5}\right) + 9(3) = \frac{21}{5} + 27$$ Convert 27 to a fraction: $$\frac{21}{5} + \frac{135}{5} = \frac{156}{5} \quad (\text{not equal to } 37)$$ Hence, it is **not** a solution. Would you like more detailed explanations on any step, or have further questions? Here are five related questions: 1. How do you check if an ordered pair is a solution to a system of equations? 2. Can systems of linear equations have no solution? 3. What does it mean geometrically if two equations don't share a solution? 4. What is the method of substitution in solving linear systems? 5. How can matrix multiplication be used to solve systems of equations? **Tip**: Always substitute carefully to avoid sign errors, especially when dealing with fractions and matrices.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems
Substitution Method
Fractions
Formulas
For the system of equations: 2x - 3y = -3
For the second equation: 7x + 9y = 37
Theorems
Method of substitution: Substituting the values of x and y into the system of equations to verify solutions.
Suitable Grade Level
Grades 8-10