Find an equation of a line passing through the point that is the solution of \(x + y = 2\) and \(2x - y = 1\). How many such lines are there?
Solution point is \((1,1)\). Infinitely many lines pass through it; e.g., \(y - 1 = m(x - 1)\).
Step 1: We are given two equations:
\(x + y = 2\)
\(2x - y = 1\)
Step 2: From the first equation:
\(x + y = 2\)
So, \(y = 2 - x\).
Step 3: Put this value of \(y\) into the second equation:
\(2x - y = 1\)
\(2x - (2 - x) = 1\)
Step 4: Simplify:
\(2x - 2 + x = 1\)
\(3x - 2 = 1\)
Step 5: Add 2 to both sides:
\(3x = 3\)
Step 6: Divide by 3:
\(x = 1\)
Step 7: Now, put \(x = 1\) into \(y = 2 - x\):
\(y = 2 - 1 = 1\)
So, the solution point is \((1,1)\).
Step 8: A line through this point can be written in slope form:
\(y - 1 = m(x - 1)\)
Here, \(m\) is the slope and can be any real number.
Final Result: Infinitely many lines can pass through the point \((1,1)\).