Math Problem Statement

Solve the system of equations: 3x = 2y - 2 and 5x = -4(15 + 2y).

Solution

The image contains a system of two linear equations that we need to solve:

3x = 2y - 2 \\ 5x = -4(15 + 2y) \end{cases}$$ ### Step 1: Simplify the second equation First, let's expand and simplify the second equation: $$5x = -4(15 + 2y) \implies 5x = -60 - 8y$$ Thus, the second equation becomes: $$5x + 8y = -60$$ ### Step 2: Solve the first equation for $$x$$ From the first equation, solve for $$x$$ in terms of $$y$$: $$3x = 2y - 2 \implies x = \frac{2y - 2}{3}$$ ### Step 3: Substitute $$x$$ into the second equation Now, substitute this expression for $$x$$ into the second equation: $$5\left(\frac{2y - 2}{3}\right) + 8y = -60$$ Multiply through by 5: $$\frac{5(2y - 2)}{3} + 8y = -60 \implies \frac{10y - 10}{3} + 8y = -60$$ Multiply the entire equation by 3 to eliminate the fraction: $$10y - 10 + 24y = -180$$ Combine like terms: $$34y - 10 = -180$$ Solve for $$y$$: $$34y = -170 \implies y = \frac{-170}{34} = -5$$ ### Step 4: Solve for $$x$$ Now substitute $$y = -5$$ into the expression for $$x$$: $$x = \frac{2(-5) - 2}{3} = \frac{-10 - 2}{3} = \frac{-12}{3} = -4$$ ### Final solution: The solution to the system of equations is: $$x = -4, \quad y = -5$$ Would you like further details or have any questions? --- Here are five related questions to explore: 1. How would you graphically represent the system of equations? 2. What does it mean if two linear equations have no solution? 3. How can you use matrices to solve a system of equations? 4. What are the advantages of using substitution over elimination in solving systems? 5. How does the system change if the second equation had been different, for example, $$5x = -4(10 + 2y)$$? **Tip:** Always check your solution by substituting the values of $$x$$ and $$y$$ back into the original equations to verify accuracy!

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Substitution Method
Simplification

Formulas

3x = 2y - 2
5x = -4(15 + 2y)
x = (2y - 2)/3

Theorems

Substitution method for solving systems of linear equations

Suitable Grade Level

Grades 8-10

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