solve by addition method 5x+3y=1 and 3x+4y=-6

We are given the system of equations:

1. \( 5x + 3y = 1 \)
2. \( 3x + 4y = -6 \)

To solve this using the addition (or elimination) method, we'll eliminate one variable by making the coefficients of one variable the same in both equations.

### Step 1: Eliminate \( x \)

We can eliminate \( x \) by finding a common multiple of the coefficients of \( x \) (5 and 3). The least common multiple of 5 and 3 is 15.

- Multiply the first equation by 3:
  \[
  3(5x + 3y) = 3(1) \quad \Rightarrow \quad 15x + 9y = 3
  \]
  
- Multiply the second equation by 5:
  \[
  5(3x + 4y) = 5(-6) \quad \Rightarrow \quad 15x + 20y = -30
  \]

Now we have the system:
\[
15x + 9y = 3
\]
\[
15x + 20y = -30
\]

### Step 2: Subtract the equations

Now subtract the first equation from the second to eliminate \( x \):

\[
(15x + 20y) - (15x + 9y) = -30 - 3
\]

Simplifying:
\[
11y = -33
\]

### Step 3: Solve for \( y \)

\[
y = \frac{-33}{11} = -3
\]

### Step 4: Substitute \( y \) into one of the original equations

Now that we have \( y = -3 \), substitute this value into one of the original equations, say \( 5x + 3y = 1 \):

\[
5x + 3(-3) = 1
\]

Simplify:
\[
5x - 9 = 1
\]

Add 9 to both sides:
\[
5x = 10
\]

Solve for \( x \):
\[
x = \frac{10}{5} = 2
\]

### Final Solution

The solution to the system is:
\[
x = 2, \quad y = -3
\]

Would you like further details or clarifications? Here are some related questions:

1. How would you solve this system using substitution instead of addition?
2. Can a system of linear equations have no solutions or infinite solutions? How would you determine that?
3. What is the geometric interpretation of solving a system of linear equations?
4. How can you check the solution for consistency in both equations?
5. How do you solve a system of three linear equations in three variables?

**Tip:** Always check your solution by substituting the values back into the original equations to verify they are correct.

Learn how to solve the system of linear equations 5x + 3y = 1 and 3x + 4y = -6 using the addition method. This step-by-step guide covers key algebraic techniques like elimination, substitution, and solving for variables. Suitable for students in Grades 9-12.

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