the line l:3x-5y+15=0 and the line m:3x+4y-12=0 cut the x-axis at the points c and d, respectively. Find e, the point of intersection of l and m.

To find the point of intersection \(E\) of the two lines \(l\) and \(m\), we need to solve the system of linear equations given by the equations of the lines.

The equations of the lines are:
\[
l: 3x - 5y + 15 = 0
\]
\[
m: 3x + 4y - 12 = 0
\]

### Step 1: Solve the system of equations

To solve for \(x\) and \(y\), we can use the method of elimination or substitution. Let's use elimination in this case.

Subtract the equation of \(m\) from the equation of \(l\):

\[
(3x - 5y + 15) - (3x + 4y - 12) = 0
\]

This simplifies to:

\[
-9y + 27 = 0
\]

Solve for \(y\):

\[
-9y = -27
\]
\[
y = 3
\]

### Step 2: Substitute \(y\) back into one of the original equations

Now that we have \(y = 3\), we can substitute it back into one of the original equations to find \(x\). Let's use the equation of line \(l\):

\[
3x - 5(3) + 15 = 0
\]

Simplify:

\[
3x - 15 + 15 = 0
\]
\[
3x = 0
\]
\[
x = 0
\]

### Step 3: Write down the point of intersection \(E\)

The point of intersection \(E\) is:
\[
E = (0, 3)
\]

This is the point where the two lines \(l\) and \(m\) intersect.

Would you like further details on this solution or have any additional questions?

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Here are 5 related questions to expand your understanding:

1. What are the coordinates of the points \(C\) and \(D\) where the lines \(l\) and \(m\) intersect the x-axis?
2. How would you find the equation of a line perpendicular to \(l\) passing through \(E\)?
3. What is the area of the triangle formed by the points \(C\), \(D\), and \(E\)?
4. How can you determine if the lines \(l\) and \(m\) are parallel or perpendicular?
5. Can you find the distance between the point \(E\) and the origin?

**Tip:** When solving systems of linear equations, it can be helpful to first simplify the equations as much as possible before using substitution or elimination.

The line l: 3x - 5y + 15 = 0 and the line m: 3x + 4y - 12 = 0 cut the x-axis at the points C and D, respectively. Find E, the point of intersection of l and m.

Learn how to find the point of intersection for two lines given by 3x - 5y + 15 = 0 and 3x + 4y - 12 = 0 using the elimination method. The solution includes solving the system of linear equations step-by-step. Suitable for students in Grades 9-10.

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