Math Problem Statement

2x-5y=3. 3x+2y=14

Solution

We can solve this system of linear equations using either substitution or elimination. I'll use the elimination method for this example:

The given system is:

(1) \quad 2x - 5y &= 3 \\ (2) \quad 3x + 2y &= 14 \end{align*}$$ ### Step 1: Eliminate one variable To eliminate one variable, let's make the coefficients of $$y$$ in both equations the same. We can do this by multiplying equation (1) by 2 and equation (2) by 5: $$\begin{align*} 2(2x - 5y) &= 2(3) \quad \Rightarrow \quad 4x - 10y = 6 \quad \text{(3)} \\ 5(3x + 2y) &= 5(14) \quad \Rightarrow \quad 15x + 10y = 70 \quad \text{(4)} \end{align*}$$ ### Step 2: Add the two equations Now, add equation (3) and equation (4) to eliminate $$y$$: $$(4x - 10y) + (15x + 10y) = 6 + 70$$ Simplifying: $$19x = 76$$ ### Step 3: Solve for $$x$$ Divide both sides by 19: $$x = \frac{76}{19} = 4$$ ### Step 4: Substitute $$x = 4$$ into one of the original equations Now substitute $$x = 4$$ into equation (1): $$2(4) - 5y = 3$$ This simplifies to: $$8 - 5y = 3$$ Solving for $$y$$: $$-5y = 3 - 8 = -5$$ $$y = \frac{-5}{-5} = 1$$ ### Final Answer: The solution to the system is $$x = 4$$ and $$y = 1$$. Would you like more details or have any questions? Here are 5 related questions: 1. How can the substitution method be used to solve the same system? 2. What happens if both equations have the same slope in a system of linear equations? 3. How can graphing be used to solve systems of linear equations? 4. What is the determinant of a 2x2 system and how does it relate to solving it? 5. How do you know if a system of equations has no solution or infinitely many solutions? **Tip:** When solving a system of equations, always check the solution by substituting the values back into both original equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations
Elimination Method

Formulas

Elimination method: Adding or subtracting equations to eliminate one variable

Theorems

Fundamental Theorem of Algebra: Every system of linear equations has either one solution, no solution, or infinitely many solutions

Suitable Grade Level

Grades 8-10

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