Math Problem Statement
Solve the system of equations: x/2 - y/3 = 2 and x/4 + y/2 = 3.
Solution
The problem asks for the solution set of the following system of equations:
\frac{x}{2} - \frac{y}{3} = 2 \\ \frac{x}{4} + \frac{y}{2} = 3 \end{cases}$$ Let's solve this system step-by-step. ### Step 1: Solve the first equation The first equation is: $$\frac{x}{2} - \frac{y}{3} = 2$$ Multiply the entire equation by 6 to eliminate the denominators: $$6 \times \left(\frac{x}{2} - \frac{y}{3}\right) = 6 \times 2$$ This simplifies to: $$3x - 2y = 12$$ So, the first equation becomes: $$3x - 2y = 12 \quad (1)$$ ### Step 2: Solve the second equation The second equation is: $$\frac{x}{4} + \frac{y}{2} = 3$$ Multiply the entire equation by 4 to eliminate the denominators: $$4 \times \left(\frac{x}{4} + \frac{y}{2}\right) = 4 \times 3$$ This simplifies to: $$x + 2y = 12$$ So, the second equation becomes: $$x + 2y = 12 \quad (2)$$ ### Step 3: Solve the system Now, we have the system: $$\begin{cases} 3x - 2y = 12 \\ x + 2y = 12 \end{cases}$$ Add both equations to eliminate $$y$$: $$(3x - 2y) + (x + 2y) = 12 + 12$$ This simplifies to: $$4x = 24$$ So, $$x = 6$$ ### Step 4: Substitute $$x = 6$$ into the second equation Substitute $$x = 6$$ into equation (2): $$6 + 2y = 12$$ This simplifies to: $$2y = 6 \quad \Rightarrow \quad y = 3$$ ### Solution The solution set is: $$(x, y) = (6, 3)$$ --- Do you have any questions, or would you like further details on any step? Here are 5 related questions you can explore: 1. How can we solve systems of equations using substitution? 2. What is the method of elimination for solving linear systems? 3. How can you solve systems of equations with three variables? 4. What are consistent, inconsistent, and dependent systems? 5. How can matrices be used to solve systems of equations? **Tip**: Multiplying both sides of an equation by a common denominator helps eliminate fractions and simplifies solving equations.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Equations
Simultaneous Equations
Algebra
Formulas
Elimination Method
Substitution Method
Theorems
System of Linear Equations
Suitable Grade Level
Grades 9-12
Related Recommendation
Solving System of Linear Equations with Fractions
Solve the System of Linear Equations 3x + 2y = 3 and 2x + 3y = 2 Using Elimination
Solving a System of Linear Equations with Fractions: 2x/3 - y/2 = -1/6 and x/2 + 2y/3 = 3
Solve a System of Equations Using Elimination Method: 4x - (1/3)y = 2/3 and (1/2)x - (1/5)y = -11
Solving a System of Linear Equations with Fractions using Elimination