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

Given equations:

1. $-15x + 6y = 18$ $\quad \text{(Equation 1)}$
2. $2y = 5x + 6$ $\quad \text{(Equation 2)}$

Step 1: Rearrange Equation 2

First, we rewrite Equation 2 to match the format of Equation 1 by moving all terms to one side:

$2y - 5x = 6 \quad \text{(Equation 2, rearranged)}$

Step 2: Align the Equations

Now, we align the two equations:

1. $-15x + 6y = 18$
2. $-5x + 2y = 6$

Step 3: Make the Coefficients of $x$ or $y$ Equal

To eliminate a variable, we want the coefficients of either $x$ or $y$ to be the same (with opposite signs). Let's eliminate $x$.

Multiply Equation 2 by 3 to make the coefficients of $x$ the same:

$3(-5x + 2y) = 3(6)$

This gives:

$-15x + 6y = 18 \quad \text{(Equation 2 multiplied by 3)}$

Step 4: Subtract the Equations

Now subtract the new Equation 2 from Equation 1:

$(-15x + 6y) - (-15x + 6y) = 18 - 18$

This simplifies to:

$0 = 0$

Conclusion:

The result $0 = 0$ indicates that the two equations are actually the same line. Thus, the system has infinitely many solutions, meaning the two lines coincide.

Would you like more details or have any questions?

Related Questions:

1. How do we graphically interpret a system with infinitely many solutions?
2. What does it mean when two equations represent the same line?
3. How can we find a specific solution to this system?
4. How can we solve a system with different slopes using elimination?
5. What are other methods to solve systems of equations, such as substitution or matrix methods?

Tip: Always check if the system of equations represents parallel lines, the same line, or intersecting lines to determine the number of solutions.