Solve the system of equations. 2x – y = 7 x + y = 2
The Correct Answer and Explanation is:
To solve the system of equations:
- 2x−y=72x – y = 72x−y=7
- x+y=2x + y = 2x+y=2
We can use either the substitution method or the elimination method. Let’s solve it step by step using the substitution method.
Step 1: Solve one equation for one variable
We’ll start by solving the second equation for yyy:x+y=2x + y = 2x+y=2
Solving for yyy, we get:y=2−xy = 2 – xy=2−x
Step 2: Substitute the expression for yyy into the first equation
Now, substitute y=2−xy = 2 – xy=2−x into the first equation 2x−y=72x – y = 72x−y=7:2x−(2−x)=72x – (2 – x) = 72x−(2−x)=7
Step 3: Simplify the equation
Distribute the negative sign:2x−2+x=72x – 2 + x = 72x−2+x=7
Combine like terms:3x−2=73x – 2 = 73x−2=7
Step 4: Solve for xxx
Now, add 2 to both sides to isolate the term with xxx:3x=93x = 93x=9
Finally, divide by 3:x=3x = 3x=3
Step 5: Solve for yyy
Now that we have x=3x = 3x=3, substitute this value back into the equation y=2−xy = 2 – xy=2−x:y=2−3=−1y = 2 – 3 = -1y=2−3=−1
Final Answer
The solution to the system of equations is:x=3,y=−1x = 3, \quad y = -1x=3,y=−1
Verification:
To verify, substitute x=3x = 3x=3 and y=−1y = -1y=−1 into the original equations:
- 2x−y=72x – y = 72x−y=7 becomes 2(3)−(−1)=6+1=72(3) – (-1) = 6 + 1 = 72(3)−(−1)=6+1=7, which is correct.
- x+y=2x + y = 2x+y=2 becomes 3+(−1)=23 + (-1) = 23+(−1)=2, which is also correct.
Therefore, the solution x=3x = 3x=3 and y=−1y = -1y=−1 is correct!
