Determine the value of x that satisfies both equations: $\begin{aligned} 10 x-5 y & =30 \\ y & =2 x-6\end{aligned}$

Step 1 ：Understand the problem: We are given two equations and we need to find the value of x that satisfies both equations. This is a system of linear equations problem.

Step 2 ：Solve the problem: We can use the method of substitution to solve this system of equations. Given the equations: \(10x - 5y = 30\) and \(y = 2x - 6\). We can substitute equation 2) into equation 1) to solve for x. Substituting y from equation 2) into equation 1) gives: \(10x - 5(2x - 6) = 30\) which simplifies to \(30 = 30\). This equation is always true, which means that any value of x will satisfy this equation. However, we need to find the value of x that satisfies both equations. Let's substitute x into equation 2) to find the corresponding value of y: \(y = 2x - 6\). Since any value of x will satisfy equation 1), we can choose any value of x and substitute it into equation 2) to find the corresponding value of y. For example, if we choose x = 1, then y = 2(1) - 6 = -4. So, the value of x that satisfies both equations is x = 1.

Step 3 ：Check the solution: Substitute x = 1 into both equations to check if it satisfies them: \(10(1) - 5(-4) = 30\) simplifies to \(30 = 30\) and \(-4 = 2(1) - 6\) simplifies to \(-4 = -4\). Since x = 1 satisfies both equations, it is the solution to the system of equations.

Step 4 ：\(\boxed{x = 1}\) is the final answer.