Math Problem Statement

Solve the system of equations: \( \frac{x+3}{3} + \frac{y}{4} = 4 \) and \( \frac{x}{6} + \frac{y+1}{5} = 2 \).

Solution

The given system of equations is:

\frac{x+3}{3} + \frac{y}{4} = 4 \\ \frac{x}{6} + \frac{y+1}{5} = 2 \end{cases}$$ Let's solve this system step by step. --- ### Step 1: Rewrite the equations Rewrite each equation to eliminate fractions. 1. Multiply the first equation by the least common denominator (LCD), which is $$12$$: $$12 \cdot \left(\frac{x+3}{3} + \frac{y}{4}\right) = 12 \cdot 4$$ $$4(x+3) + 3y = 48$$ Simplify: $$4x + 12 + 3y = 48 \quad \Rightarrow \quad 4x + 3y = 36 \tag{1}$$ 2. Multiply the second equation by the LCD, which is $$30$$: $$30 \cdot \left(\frac{x}{6} + \frac{y+1}{5}\right) = 30 \cdot 2$$ $$5x + 6(y+1) = 60$$ Simplify: $$5x + 6y + 6 = 60 \quad \Rightarrow \quad 5x + 6y = 54 \tag{2}$$ --- ### Step 2: Solve the system of linear equations The simplified system of equations is: $$\begin{cases} 4x + 3y = 36 \tag{1} \\ 5x + 6y = 54 \tag{2} \end{cases}$$ #### Solve for $$x$$ and $$y$$: 1. Multiply Equation (1) by 2 to align the coefficients of $$y$$: $$2(4x + 3y) = 2(36) \quad \Rightarrow \quad 8x + 6y = 72 \tag{3}$$ 2. Subtract Equation (2) from Equation (3): $$(8x + 6y) - (5x + 6y) = 72 - 54$$ $$3x = 18 \quad \Rightarrow \quad x = 6$$ 3. Substitute $$x = 6$$ into Equation (1): $$4(6) + 3y = 36$$ $$24 + 3y = 36 \quad \Rightarrow \quad 3y = 12 \quad \Rightarrow \quad y = 4$$ --- ### Final Solution: $$x = 6, \, y = 4$$ Let me know if you'd like a detailed explanation of any step! --- ### Related Questions: 1. How do you verify solutions for a system of equations? 2. What methods can be used to solve a system of linear equations with fractions? 3. How do you find the least common denominator (LCD) for multiple fractions? 4. How can elimination and substitution methods be combined in solving equations? 5. What real-world problems can be modeled using systems of linear equations? ### Tip: When solving systems of equations with fractions, always eliminate the fractions first by multiplying through by the least common denominator to simplify calculations.

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Math Problem Analysis

Mathematical Concepts

Linear equations
Systems of equations
Fractions

Formulas

Elimination method
Fraction manipulation

Theorems

Linear combination method for systems of equations

Suitable Grade Level

Grades 9-12

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