find the point of interception between the lines 3x+4y and 2x-3y=13

To find the point of intersection between the two lines:

1. The first equation is: 
\[
3x + 4y = 0
\]
2. The second equation is: 
\[
2x - 3y = 13
\]

We'll solve this system of equations using either substitution or elimination. Let's use the elimination method.

### Step 1: Eliminate one variable

We aim to eliminate \(x\) or \(y\). To do this, we multiply the first equation by 2 and the second equation by 3, so that the coefficients of \(x\) will be equal. This gives us:

\[
6x + 8y = 0 \quad \text{(from multiplying the first equation by 2)}
\]
\[
6x - 9y = 39 \quad \text{(from multiplying the second equation by 3)}
\]

### Step 2: Subtract the two equations

Now subtract the second equation from the first equation:

\[
(6x + 8y) - (6x - 9y) = 0 - 39
\]

This simplifies to:

\[
17y = -39
\]

Solving for \(y\):

\[
y = \frac{-39}{17}
\]

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

Now that we have \(y = \frac{-39}{17}\), substitute this value into one of the original equations to find \(x\). Let's use the first equation \(3x + 4y = 0\):

\[
3x + 4\left(\frac{-39}{17}\right) = 0
\]

This simplifies to:

\[
3x - \frac{156}{17} = 0
\]

\[
3x = \frac{156}{17}
\]

\[
x = \frac{156}{51} = \frac{52}{17}
\]

### Final Answer:
The point of intersection is:

\[
\left( \frac{52}{17}, \frac{-39}{17} \right)
\]

Would you like more details on any step, or have any questions?

Here are 5 related questions you can explore:
1. What is the geometric meaning of the intersection point?
2. How do you solve systems of equations graphically?
3. Can you apply substitution to solve the same system?
4. How can you solve non-linear systems of equations?
5. What are some applications of systems of linear equations in real life?

**Tip**: When solving systems of equations, always check if substitution or elimination is easier based on the coefficients.

Learn how to find the point of intersection between the lines 3x + 4y = 0 and 2x - 3y = 13 using the elimination method. This step-by-step solution is ideal for students in Grades 9-12 and covers key algebraic concepts and methods for solving systems of linear equations.

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