Question

How do you write the ordered pair that is the solution to the following system of equations: \[y = 2x + 1\] and \[y = x - 5\] ?

Answer 1

Hint: In this question you have to find the ordered pair that is \[\left( {x,y} \right)\] of the given two equations by two different methods, firstly by using method of substitution otherwise by using the elimination method of simultaneous equation. In elimination methods either you add or subtract the equations.

Complete step-by-step answer:
The ordered pair is simply the point at which these two equations of lines cross.
solve the equations as simultaneous equations using the method of substitution
consider the given equations
 \[y = 2x + 1\] -------(1)
 \[y = x - 5\] --------(2)
Rearrange the equation (2) to make \[x\] the subject
 \[x = y + 5\]
Substitute the value of \[x\] in equation (1) , then
 \[y = 2\left( {y + 5} \right) + 1\]
 \[y = 2y + 10 + 1\]
 \[y = 2y + 11\]
Take all \[y\] terms to LHS
 \[y - 2y = 11\]
 \[ - y = 11\]
 \[\therefore \,\,\,y = - 11\]
Substitute the \[y\] value in equation of \[x\]
 \[x = - 11 + 5\]
 \[\therefore \,\,\,x = - 6\]
Or
Since both equations express \[y\] in terms of \[x\] , you can equate to each other, then
 \[ \Rightarrow 2x + 1 = x - 5\]
Subtract \[x\] from both sides
 \[ \Rightarrow 2x + 1 - x = x - x - 5\]
 \[ \Rightarrow x + 1 = - 5\]
 \[ \Rightarrow x = - 5 - 1\]
 \[\therefore \,\,\,x = - 6\]
Substitute the \[x\] value in equation (1) to get value \[y\] .
 \[ \Rightarrow \,y = 2\left( { - 6} \right) + 1\]
 \[\therefore y = - 11\]
So, the correct answer is “x = -6 AND y = -11”.

Note: If a, b, and r are real numbers (and if a and b are not both equal to 0) then ax+by = r is called a linear equation in two variables. (The “two variables” are the x and the y.) The numbers a and b are called the coefficients of the equation ax+by = r. The number r is called the constant of the equation ax + by = r