3x - 4y = -5
9x +5y = -4
solve by elimination 

To solve the system of linear equations using the elimination method, we follow these steps:

Given equations:

1. \(3x - 4y = -5\)  \(\quad \text{(Equation 1)}\)
2. \(9x + 5y = -4\)  \(\quad \text{(Equation 2)}\)

### Step 1: Make the Coefficients of One Variable Equal
We aim to eliminate one variable by making the coefficients of either \(x\) or \(y\) equal (with opposite signs). Let's eliminate \(x\) first. 

To do this, we'll multiply Equation 1 by 3 to match the coefficient of \(x\) in Equation 2.

\[
3(3x - 4y) = 3(-5)
\]

This gives:

\[
9x - 12y = -15 \quad \text{(Equation 1 multiplied by 3)}
\]

Now we have:

1. \(9x - 12y = -15\)  \(\quad \text{(Equation 3)}\)
2. \(9x + 5y = -4\)  \(\quad \text{(Equation 2)}\)

### Step 2: Subtract the Equations to Eliminate \(x\)
Next, subtract Equation 2 from Equation 3 to eliminate \(x\):

\[
(9x - 12y) - (9x + 5y) = -15 - (-4)
\]

Simplifying this, we get:

\[
9x - 12y - 9x - 5y = -15 + 4
\]

\[
-17y = -11
\]

### Step 3: Solve for \(y\)
Divide both sides by \(-17\) to solve for \(y\):

\[
y = \frac{-11}{-17} = \frac{11}{17}
\]

### Step 4: Substitute \(y\) Back into One of the Original Equations
Now, substitute \(y = \frac{11}{17}\) back into one of the original equations to solve for \(x\). We'll use Equation 1:

\[
3x - 4\left(\frac{11}{17}\right) = -5
\]

Multiply:

\[
3x - \frac{44}{17} = -5
\]

Next, add \(\frac{44}{17}\) to both sides:

\[
3x = -5 + \frac{44}{17}
\]

To combine the terms on the right, write \(-5\) as \(\frac{-85}{17}\):

\[
3x = \frac{-85 + 44}{17} = \frac{-41}{17}
\]

Now, divide both sides by 3:

\[
x = \frac{-41}{17} \times \frac{1}{3} = \frac{-41}{51}
\]

### Final Answer:
The solution to the system is:

\[
x = \frac{-41}{51}, \quad y = \frac{11}{17}
\]

Would you like more details or have any questions?

### Related Questions:
1. How can you verify the solution of a system of equations?
2. What are the advantages of using elimination over substitution?
3. How do you solve a system of equations when both variables have fractional coefficients?
4. What happens if elimination leads to a false statement like \(0 = 1\)?
5. How can we use matrices to solve a system of linear equations?

**Tip:** Always simplify your equations as much as possible before solving them, which can reduce calculation errors and make the process more manageable.

Learn how to solve the system of linear equations 3x - 4y = -5 and 9x + 5y = -4 using the elimination method with a step-by-step solution. This problem involves eliminating variables to find the values of x and y. Suitable for high school students in Grades 9-12.

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