8. Solve the following system of linear equations: $3x + y = 10$ $x - y = 2$ (system of linear equations, elimination method, substitution method)

Answer: The solution to the system of equations is $x = 3$ and $y = 1$ .

Step-by-step solution

Step 1: Identify the equations

We are given two linear equations. Let's label them as Equation (1) and Equation (2) for easy reference. Our goal is to find the values of $x$ and $y$ that satisfy both equations simultaneously.

Step 2: Eliminate 'y' by adding the equations

Notice that the coefficients of $y$ in the two equations are $+1$ and $-1$ . By adding Equation (1) and Equation (2), the $y$ terms will cancel out, allowing us to solve for $x$ . This is known as the elimination method.

Step 3: Solve for 'x'

Now that we have $4x = 12$ , we can isolate $x$ by dividing both sides of the equation by 4. This gives us the value of $x$ .

Step 4: Substitute 'x' into one of the original equations

Now that we know $x = 3$ , we can substitute this value into either Equation (1) or Equation (2) to find $y$ . Let's use Equation (1) for this step.

Step 5: Solve for 'y'

After substituting $x=3$ into Equation (1), we simplify the equation to $9 + y = 10$ . Subtracting 9 from both sides gives us the value of $y$ .

Step 6: Verify the solution

To ensure our solution is correct, we substitute $x=3$ and $y=1$ into both original equations. If both equations hold true, our solution is correct.

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8. Solve the following system of linear equations: 3x+y=103x + y = 103x+y=10 x−y=2x - y = 2x−y=2 (system of linear equations, elimination method, substitution method)