To solve the given system of linear equations:

$-5x + 3y = 0$ $7x - 4y = 2$

we aim to eliminate one variable by combining the equations. Here's the step-by-step process:

Step 1: Align equations

$-5x + 3y = 0 \quad \text{(Equation 1)}$ $7x - 4y = 2 \quad \text{(Equation 2)}$

Step 2: Eliminate $y$

To eliminate $y$, multiply both equations by suitable constants so that the coefficients of $y$ in both equations are equal in magnitude.

Multiply Equation 1 by 4 and Equation 2 by 3: $4(-5x + 3y) = 4(0) \implies -20x + 12y = 0$ $3(7x - 4y) = 3(2) \implies 21x - 12y = 6$

Now the system becomes: $-20x + 12y = 0$ $21x - 12y = 6$

Step 3: Add the equations to eliminate $y$

$(-20x + 12y) + (21x - 12y) = 0 + 6$ $x = 6$

Step 4: Substitute $x = 6$ into one of the original equations

Substitute $x = 6$ into Equation 1: $-5(6) + 3y = 0$ $-30 + 3y = 0$ $3y = 30$ $y = 10$

Final Solution:

The solution to the system is: $x = 6, \, y = 10$

Would you like additional details, a verification step, or to explore similar examples?

Related Questions:

1. How can we use substitution instead of elimination to solve this system?
2. What happens if we multiply both equations differently in the elimination method?
3. How do we graphically interpret the solution of a system of linear equations?
4. What are the steps to confirm that $x = 6, y = 10$ satisfies both equations?
5. Can we solve this system using matrix methods or determinants?

Tip:

When using the elimination method, always ensure the coefficients of one variable match in magnitude before adding or subtracting the equations.