Solving Systems of Linear Equations Using the Echelon Method

Math Problem Statement

Use the echelon method to solve the following system of two equations in two unknowns. 3x + 2y = -9 and 5x - 2y = -15.

Let's solve the system of equations using the echelon method.

The given system is:

1. \quad & 3x + 2y = -9 \\ 2. \quad & 5x - 2y = -15 \end{aligned}$$ ### Step 1: Eliminate one variable To eliminate one variable, we can add the two equations to cancel out the $$y$$-terms. Add equation (1) and (2): $$(3x + 2y) + (5x - 2y) = -9 + (-15)$$ This simplifies to: $$8x = -24$$ Solving for $$x$$: $$x = \frac{-24}{8} = -3$$ ### Step 2: Solve for $$y$$ Now substitute $$x = -3$$ into one of the original equations. Let's use equation (1): $$3(-3) + 2y = -9$$ Simplifying: $$-9 + 2y = -9$$ Add 9 to both sides: $$2y = 0$$ Solve for $$y$$: $$y = 0$$ ### Final Answer: The solution is $$(x, y) = (-3, 0)$$. Therefore, the correct choice is: - **A. The solution is (-3, 0)**. --- Would you like further details or have any questions? Here are some related questions to consider: 1. How does the echelon method differ from substitution or elimination methods? 2. What other methods can be used to solve systems of equations? 3. How would the solution change if both equations were parallel? 4. How can systems of equations be applied in real-life scenarios? 5. What is the significance of having infinitely many solutions? **Tip**: When solving a system of equations, always check your solution by substituting the values back into the original equations to ensure accuracy.

Drop file here or Click Here to upload

Linear Algebra
Systems of Equations
Echelon Method

Standard form of a linear equation: ax + by = c
Row reduction methods

Echelon Form

Grades 9-12

Solving a System of Equations Using the Echelon Method: 7x − 6y = −55 and −14x + 12y = 10

Solving Systems of Equations using Echelon Method: 2x + 3y = -2 and 3x - 2y = 10

Solve System of Equations Using Echelon Method: 6x + 3y = -12, 5x - 3y = -21

Solving a System of Linear Equations using the Echelon Method

Solving System of Linear Equations -2x - 5y = -15 and -9x + 15y = -30