Solve the pair of linear equations: x + y=14 and x - y=4.
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Step-by-Step Solution
Step 1: Write down the given equations
We are given two linear equations with two variables, x and y. We need to find the values of x and y that satisfy both equations simultaneously.
Step 2: Add the two equations
We can eliminate one of the variables by adding the two equations. Notice that the y terms have opposite signs (+y and −y), so they will cancel out when added. Adding the left sides gives (x+y)+(x−y)=2x, and adding the right sides gives 14+4=18.
Step 3: Solve for x
Now we have a simple linear equation with only one variable, x. To find the value of x, we divide both sides of the equation by 2. This gives us x=9.
Step 4: Substitute x into one of the original equations
We now know the value of x. We can substitute this value into either Equation 1 or Equation 2 to find y. Let's use Equation 1. Substituting x=9 into x+y=14 gives us 9+y=14.
Step 5: Solve for y
To solve for y, subtract 9 from both sides of the equation. This gives us y=5. We have now found the values for both x and y.
Step 6: Verify the solution
It's always a good practice to verify the solution by substituting the values of x and y back into both original equations. For Equation 1, 9+5=14, which is correct. For Equation 2, 9−5=4, which is also correct. Thus, our solution is verified.