Question

Solve the system of linear equations \[y = - 2x + 5\] and \[y = - 2x + 20\]?

Answer 1

Hint: We use the substitution method to solve two linear equations given in the question. We find the value of y from the first equation in terms of x and substitute in the second equation which becomes an equation in x entirely. Solve for the value of x and substitute back the value of y to obtain the value of y.

Complete step by step solution:
We have two linear equations \[y = - 2x + 5\] and \[y = - 2x + 20\]
Let us solve the first equation to obtain the value of y in terms of x.
We have \[y = - 2x + 5\] … (1)
Now we substitute the value of \[y = - 2x + 5\] from equation (1) in the second linear equation.
Substitute \[y = - 2x + 5\] in \[y = - 2x + 20\]
\[ \Rightarrow - 2x + 5 = - 2x + 20\]
Bring all variables to one side and all constants to other side of the equation
\[ \Rightarrow - 2x + 2x = 20 - 5\]
\[ \Rightarrow 0 = 15\]
This is a contradiction as 0 is not equal to 15.
So, the system of linear equations has no solution.
\[\therefore \]System of linear equations \[y = - 2x + 5\] and \[y = - 2x + 20\] has no solution.

Note: Alternate method:
We know the equation of the line is given by \[y = mx + c\], where m is the slope of the line.
Compare both the equations of lines \[y = - 2x + 5\] and \[y = - 2x + 20\] with the general equation of line.
We observe that the slope of both the lines is -2. Also, we know that the slope of any two lines is equal when the two lines are parallel to each other.
So, the lines \[y = - 2x + 5\] and \[y = - 2x + 20\] are parallel to each other
Therefore, we can say lines \[y = - 2x + 5\] and \[y = - 2x + 20\] will never intersect each other and hence there will be no solution of the system of linear equations. (A solution of a system of linear equations means a point where the lines intersect each other).