Solving a System of Equations Using the Addition (Elimination) Method

Q1. Solve the system of equations by the addition method:

$\begin{cases} 3x - 9y = 15 \\ 5x - 15y = 25 \end{cases}$

Background

Topic: Systems of Linear Equations (Addition/Elimination Method)

This question tests your ability to solve a system of two linear equations using the addition (elimination) method. The goal is to eliminate one variable by adding or subtracting the equations, making it possible to solve for the other variable.

Key Terms and Formulas

• System of equations: Two or more equations with the same variables.

• Addition (Elimination) Method: A method for solving systems by adding or subtracting equations to eliminate one variable.

Step-by-Step Guidance

1. Write the system clearly: $\begin{cases} 3x - 9y = 15 \\ 5x - 15y = 25 \end{cases}$

2. Decide which variable to eliminate. Let's try to eliminate $x$ or $y$ by making the coefficients of one variable the same (or opposites) in both equations.

3. To eliminate $x$, find the least common multiple (LCM) of the coefficients of $x$ (3 and 5). The LCM is 15. Multiply the first equation by 5 and the second equation by 3 to align the $x$ coefficients:

   $5(3x - 9y) = 5(15)$ $3(5x - 15y) = 3(25)$

4. Write the new system after multiplication: $15x - 45y = 75$ $15x - 45y = 75$

5. Subtract one equation from the other to eliminate $x$ and $y$:

   $(15x - 45y) - (15x - 45y) = 75 - 75$

Try solving on your own before revealing the answer!