Math Problem Statement

Use the Addition Method to solve. 2x-y=1 3x+2y=12

Solution

To solve the system of equations using the addition method (also known as the elimination method), we aim to eliminate one variable by adding the equations together. Here are the steps:

System of equations:

1. & \quad 2x - y = 1 \\ 2. & \quad 3x + 2y = 12 \end{align*}$$ ### Step 1: Align the equations and multiply (if necessary) to eliminate one variable. We want to eliminate $$y$$. The coefficients of $$y$$ in the two equations are $$-1$$ (from $$2x - y = 1$$) and $$2$$ (from $$3x + 2y = 12$$). To eliminate $$y$$, we can multiply the first equation by 2. $$2(2x - y) = 2(1) \quad \Rightarrow \quad 4x - 2y = 2$$ Now, the system of equations is: $$\begin{align*} 1'. & \quad 4x - 2y = 2 \\ 2. & \quad 3x + 2y = 12 \end{align*}$$ ### Step 2: Add the two equations to eliminate $$y$$. Now add the two equations: $$(4x - 2y) + (3x + 2y) = 2 + 12$$ Simplifying: $$4x + 3x = 14 \quad \Rightarrow \quad 7x = 14$$ ### Step 3: Solve for $$x$$. $$x = \frac{14}{7} = 2$$ ### Step 4: Substitute $$x = 2$$ back into one of the original equations. Let's substitute $$x = 2$$ into the first equation $$2x - y = 1$$. $$2(2) - y = 1 \quad \Rightarrow \quad 4 - y = 1$$ Solving for $$y$$: $$-y = 1 - 4 \quad \Rightarrow \quad -y = -3 \quad \Rightarrow \quad y = 3$$ ### Final Answer: The solution to the system is: $$x = 2, \quad y = 3$$ Would you like more details or have any questions? --- Here are 5 related questions: 1. How would you solve the system using substitution instead of the addition method? 2. What happens if both equations are already balanced to eliminate a variable? 3. How can you verify the solution $$x = 2, y = 3$$ is correct? 4. How does the addition method compare to graphing the system of equations? 5. What would happen if you tried to eliminate $$x$$ instead of $$y$$? **Tip**: Always check your solution by substituting the values back into the original equations to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
System of Linear Equations
Elimination (Addition) Method

Formulas

2x - y = 1
3x + 2y = 12
7x = 14

Theorems

Elimination Method in Systems of Equations

Suitable Grade Level

Grades 8-10

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