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