Math Problem Statement

3x - 5y = 4 and 2x + 6y = 3

Solution

You have uploaded a system of linear equations:

3x - 5y = 4 \\ 2x + 6y = 3 \end{cases}$$ Let's solve this system of equations using the substitution or elimination method. ### Step 1: Multiply the equations to align coefficients We will use the elimination method. Multiply the first equation by 2 and the second by 3, to make the coefficients of $$x$$ the same: $$\text{First equation: } 2(3x - 5y = 4) \implies 6x - 10y = 8$$ $$\text{Second equation: } 3(2x + 6y = 3) \implies 6x + 18y = 9$$ ### Step 2: Subtract the equations Now subtract the second equation from the first: $$(6x - 10y) - (6x + 18y) = 8 - 9$$ This simplifies to: $$-28y = -1$$ $$y = \frac{-1}{-28} = \frac{1}{28}$$ ### Step 3: Solve for $$x$$ Substitute $$y = \frac{1}{28}$$ into one of the original equations, say $$3x - 5y = 4$$: $$3x - 5\left(\frac{1}{28}\right) = 4$$ $$3x - \frac{5}{28} = 4$$ Multiply through by 28 to eliminate the fraction: $$28(3x) - 5 = 28(4)$$ $$84x - 5 = 112$$ $$84x = 117$$ $$x = \frac{117}{84} = \frac{39}{28}$$ ### Final solution: $$x = \frac{39}{28}, \quad y = \frac{1}{28}$$ Would you like a more detailed explanation of any step? Here are some follow-up questions to enhance understanding: 1. How do you know which method (substitution vs. elimination) is better for solving a system of equations? 2. Can you verify the solution by substituting $$x$$ and $$y$$ into both original equations? 3. How would the system change if one of the equations was non-linear? 4. Could this system have been solved graphically? What would the solution represent graphically? 5. How would you handle a system of three equations and three unknowns? **Tip:** When solving systems of equations, elimination is often preferred if the coefficients of one variable can easily be made the same.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Elimination Method

Formulas

Linear equation in two variables: ax + by = c
Elimination method for solving systems

Theorems

None required for basic linear systems

Suitable Grade Level

Grades 8-10

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