Math Problem Statement

Solve the system of equations: x + 2y = 14 and 3x - y = 7

Solution

The system of linear equations given in the image is:

x + 2y = 14 \\ 3x - y = 7 \end{cases}$$ Let's solve this system using the substitution or elimination method. I'll use substitution for this explanation: 1. From the first equation, solve for $$x$$: $$x = 14 - 2y$$ 2. Substitute $$x = 14 - 2y$$ into the second equation: $$3(14 - 2y) - y = 7$$ 3. Simplify and solve for $$y$$: $$42 - 6y - y = 7$$ $$42 - 7y = 7$$ $$-7y = 7 - 42$$ $$-7y = -35$$ $$y = 5$$ 4. Now substitute $$y = 5$$ back into the equation $$x = 14 - 2y$$: $$x = 14 - 2(5) = 14 - 10 = 4$$ Thus, the solution is: $$x = 4, \quad y = 5$$ Do you have any questions or would you like further details? Here are five related questions to expand on this: 1. What happens if you use the elimination method instead of substitution? 2. Can you represent this system graphically and find the intersection? 3. How do you verify that the solution is correct by substituting into both equations? 4. How do systems of linear equations differ from nonlinear systems? 5. How can you interpret this system in a real-world context? **Tip**: Always check your solution by substituting the values back into the original equations to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Substitution Method
Elimination Method

Formulas

Substitution Method: Solve one equation for one variable and substitute into the other.
Elimination Method: Multiply equations to align terms for elimination.

Theorems

Theorems of linear systems: If two equations are linearly independent, they have a unique solution.

Suitable Grade Level

Grades 8-10

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