Math Problem Statement
Solve the system of equations: (1/2)x + (2/3)y = 3 and (3/4)x - (1/3)y = 6.
Solution
The system of equations shown in the image is:
\frac{1}{2}x + \frac{2}{3}y = 3 \\ \frac{3}{4}x - \frac{1}{3}y = 6 \end{cases}$$ To solve this system of linear equations, we can use either the substitution method or the elimination method. Let's solve it step by step using the elimination method: ### Step 1: Eliminate the fractions by multiplying both equations by the least common denominator (LCD) of their denominators. - The LCD for the first equation is 6 (since $$2$$ and $$3$$ are the denominators). - The LCD for the second equation is 12 (since $$4$$ and $$3$$ are the denominators). For the first equation: $$6 \left( \frac{1}{2}x + \frac{2}{3}y = 3 \right) \Rightarrow 3x + 4y = 18$$ For the second equation: $$12 \left( \frac{3}{4}x - \frac{1}{3}y = 6 \right) \Rightarrow 9x - 4y = 72$$ Now the system becomes: $$\begin{cases} 3x + 4y = 18 \\ 9x - 4y = 72 \end{cases}$$ ### Step 2: Add the two equations to eliminate $$y$$. $$(3x + 4y) + (9x - 4y) = 18 + 72$$ Simplifying: $$12x = 90 \Rightarrow x = \frac{90}{12} = 7.5$$ ### Step 3: Substitute $$x = 7.5$$ into one of the original equations to solve for $$y$$. Substituting into the first equation: $$\frac{1}{2}(7.5) + \frac{2}{3}y = 3$$ Simplifying: $$3.75 + \frac{2}{3}y = 3$$ Subtracting 3.75 from both sides: $$\frac{2}{3}y = 3 - 3.75 = -0.75$$ Multiplying both sides by $$\frac{3}{2}$$: $$y = \frac{-0.75 \times 3}{2} = -1.125$$ ### Final Solution: $$x = 7.5, \quad y = -1.125$$ Let me know if you'd like more details or have questions! ### Related Questions: 1. How can we solve the system using substitution? 2. What is the graphical interpretation of this system of equations? 3. Can the elimination method always work for any system of linear equations? 4. How can we verify the solution for $$x$$ and $$y$$? 5. How would the solution change if the right-hand sides of the equations were different? ### Tip: Always check your solution by substituting both values of $$x$$ and $$y$$ into the original equations to ensure they satisfy both equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Elimination Method
Substitution Method
Formulas
Equation 1: (1/2)x + (2/3)y = 3
Equation 2: (3/4)x - (1/3)y = 6
Elimination of fractions
Adding/Subtracting equations to eliminate variables
Theorems
Systems of Linear Equations
Elimination Method
Suitable Grade Level
Grades 9-12
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