If x=3k+2 and y=2k-1 is a solution of the equation 4x-3y+1=0,find the value of k

We are given the equations:

\[
x = 3k + 2
\]
\[
y = 2k - 1
\]
and the main equation:
\[
4x - 3y + 1 = 0
\]

### Step 1: Substitute the expressions for \(x\) and \(y\) into the main equation

\[
4(3k + 2) - 3(2k - 1) + 1 = 0
\]

### Step 2: Expand the equation

\[
4(3k) + 4(2) - 3(2k) + 3(1) + 1 = 0
\]
\[
12k + 8 - 6k + 3 + 1 = 0
\]

### Step 3: Combine like terms

\[
12k - 6k + 8 + 3 + 1 = 0
\]
\[
6k + 12 = 0
\]

### Step 4: Solve for \(k\)

\[
6k = -12
\]
\[
k = -2
\]

So, the value of \(k\) is \(-2\).

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

Here are 5 related questions to explore:

1. What happens if we change the equation \(4x - 3y + 1 = 0\)?
2. How would the values of \(x\) and \(y\) change if \(k = 1\)?
3. How do we solve for multiple variables in systems of equations?
4. Can we use substitution or elimination methods for different types of equations?
5. What if we had a quadratic equation instead of a linear one?

**Tip:** Always substitute back into the original equation to check your solution!

If x=3k+2 and y=2k-1 is a solution of the equation 4x-3y+1=0, find the value of k.

Find the value of k given x=3k+2 and y=2k-1 using substitution in the equation 4x - 3y + 1 = 0. This step-by-step solution demonstrates algebraic substitution and simplification, suitable for Grades 8-10.

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