Show that lines 3x+5y=7 and 2x+3y=4 and 4x+3y=2 are concurrent.

To show that the lines \(3x + 5y = 7\), \(2x + 3y = 4\), and \(4x + 3y = 2\) are concurrent, we need to prove that all three lines intersect at a single point.

First, let's find the point of intersection of the first two lines \(3x + 5y = 7\) and \(2x + 3y = 4\).

### Step 1: Solving the system of equations

\[
\begin{cases}
3x + 5y = 7 \\
2x + 3y = 4
\end{cases}
\]

We can use the method of elimination to solve this system. First, we multiply both equations by suitable constants to make the coefficients of \(x\) or \(y\) in both equations the same. Let's eliminate \(x\) by making the coefficients of \(x\) in both equations equal.

Multiply the first equation by 2 and the second equation by 3:

\[
\begin{cases}
2(3x + 5y) = 2 \cdot 7 \\
3(2x + 3y) = 3 \cdot 4
\end{cases}
\]

This simplifies to:

\[
\begin{cases}
6x + 10y = 14 \\
6x + 9y = 12
\end{cases}
\]

Next, subtract the second equation from the first:

\[
(6x + 10y) - (6x + 9y) = 14 - 12
\]

\[
6x + 10y - 6x - 9y = 2
\]

\[
y = 2
\]

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

Let's substitute \( y = 2 \) into the first equation \( 3x + 5y = 7 \):

\[
3x + 5(2) = 7
\]

\[
3x + 10 = 7
\]

\[
3x = 7 - 10
\]

\[
3x = -3
\]

\[
x = -1
\]

Thus, the intersection point of the first two lines is \( (-1, 2) \).

### Step 3: Verify that \( (-1, 2) \) lies on the third line

Now, we need to check if this point \( (-1, 2) \) satisfies the third equation \( 4x + 3y = 2 \):

Substitute \( x = -1 \) and \( y = 2 \) into \( 4x + 3y = 2 \):

\[
4(-1) + 3(2) = 2
\]

\[
-4 + 6 = 2
\]

\[
2 = 2
\]

Since the point \( (-1, 2) \) satisfies the third equation as well, it means that all three lines intersect at the same point.

Therefore, the lines \( 3x + 5y = 7 \), \( 2x + 3y = 4 \), and \( 4x + 3y = 2 \) are concurrent at the point \( (-1, 2) \).

Would you like more details or have any questions about this explanation?

Here are 8 related questions:
1. What does it mean for lines to be concurrent?
2. How do you find the intersection point of two lines?
3. Can three lines be concurrent if they are parallel?
4. What is the method of elimination in solving systems of equations?
5. How can you verify if a point lies on a given line?
6. What is the significance of the coefficients in a linear equation?
7. Can concurrent lines be represented in three-dimensional space?
8. What is the geometric interpretation of the intersection of lines?

**Tip:** When solving systems of linear equations, always double-check your algebraic manipulations to ensure accuracy, especially when using elimination or substitution methods.

Learn how to demonstrate that the lines 3x+5y=7, 2x+3y=4, and 4x+3y=2 are concurrent with a detailed step-by-step solution. This problem involves solving systems of linear equations and verifying intersection points. Suitable for high school students studying linear algebra and geometry.

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