Math Problem Statement
Solve the system of equations by the substitution method:
x + 11y = 52
4x + 3y = 3
Solution
To solve the given system of equations by the substitution method, let's go through the steps carefully.
The system of equations is:
x + 11y = 52 \\ 4x + 3y = 3 \end{cases}$$ ### Step 1: Solve for $$x$$ in terms of $$y$$ using the first equation From the first equation: $$x = 52 - 11y$$ ### Step 2: Substitute $$x = 52 - 11y$$ into the second equation Replace $$x$$ in the second equation with $$52 - 11y$$: $$4(52 - 11y) + 3y = 3$$ ### Step 3: Simplify the equation Expanding and simplifying: $$208 - 44y + 3y = 3$$ $$208 - 41y = 3$$ $$-41y = 3 - 208$$ $$-41y = -205$$ $$y = \frac{-205}{-41} = 5$$ ### Step 4: Substitute $$y = 5$$ back into the equation for $$x$$ Now, substitute $$y = 5$$ into $$x = 52 - 11y$$: $$x = 52 - 11 \cdot 5$$ $$x = 52 - 55$$ $$x = -3$$ ### Solution The solution to the system is: $$(x, y) = (-3, 5)$$ Would you like further details or explanations on any of these steps? --- Here are some related questions to deepen understanding: 1. How can we verify that our solution is correct? 2. What would happen if we used elimination instead of substitution? 3. Are there other methods for solving systems of equations? 4. How do you handle systems with no solution or infinite solutions? 5. How can we use graphing to visualize the solution? **Tip:** When using substitution, always isolate one variable first and simplify carefully to avoid mistakes in calculations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Substitution Method
Formulas
Isolating one variable and substituting into the other equation
Theorems
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Suitable Grade Level
Grades 9-10