Use the substitution method to solve the following system of linear equations:
x + y = 6
x = –y + 2
A. There are an infinite number of solutions.
B. The solution is (2, 4).
C. The solution is (4, 2).
D. There is no solution.
since the slopes are the same, yet the lines are different, they are parallel.
(D)
To solve the system of linear equations using the substitution method, follow these steps:
Step 1: Solve one equation for one variable in terms of the other variable.
Given the second equation: x = -y + 2
We can rearrange it to isolate y: y = -x + 2.
Step 2: Substitute the expression for the variable found in Step 1 into the other equation.
Substitute y = -x + 2 into the first equation: x + (-x + 2) = 6.
Step 3: Simplify and solve the resulting equation.
Combine like terms: x - x + 2 = 6. Simplifying, we get 2 = 6. This is not true.
Step 4: Analyze the result.
Since the equation 2 = 6 is false, this means that there is no valid solution. Therefore, the correct answer is D. There is no solution.
Note: If the result had been a true equation, such as 2 = 2, then there would have been an infinite number of solutions. However, in this case, the result is false, indicating no solution.