SOLUTION: -x+3y=-6 6y=2x+6

Click here to see ALL problems on Graphs

Question 1201284: -x+3y=-6
6y=2x+6
Found 4 solutions by ikleyn, josgarithmetic, math_tutor2020, MathTherapy:
Answer by ikleyn(52884)   (Show Source):
You can put this solution on YOUR website!
.

The starting equations are -x + 3y = -6 (1) 6y = 2x + 6 (2) From (1), express 3y = -6+x and substitute it into the second equation 2*(3y) = 2x + 6, or 2*(-6+x) = 2x +6. Simplify further -12 + 2x = 2x + 6 Cancel 2x in both sides and get -12 = 6. This equality is self-contradictory, which means that the original system of equations over real numbers is INCONSISTENT and has no solution/solutions.

Solved.

-------------------

There are many other ways to get the same conclusion,
but the fact itself is indestructible as a rock.

///////////////

To learn when a system of equations is inconsistent, see the lesson
    - Geometric interpretation of the linear system of two equations in two unknowns
in this site.

Answer by josgarithmetic(39630)   (Show Source):
You can put this solution on YOUR website!

Adding corresponding members,
, but this is false.

NO SOLUTION

Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

Answer: No solutions
The system is inconsistent.
The graph is shown below (see method 3).

=============================================================================

Explanation:

There are many ways to solve a system of linear equations.
I'll go over four methods.

Method 1

Subtract 2x from both sides of the 2nd equation
6y = 2x+6
6y-2x = 6
-2x+6y = 6

The system

is equivalent to

Then notice we can divide each term of the 2nd equation by 2.
-2x+6y=6 turns into -x+3y=3

This means

is equivalent to

Let z = -x+3y represent some number

We have

update to

Uh oh. We run into a problem.
We cannot have z be equal to more than one thing at the same time.
We have a contradiction.
Therefore, this system has no solutions. We consider the system inconsistent.
It would be like flipping a coin to get heads and tails at the same time.

------------------

Method 2

Let's revisit this system

Double both sides of the 1st equation.
So we have -x+3y=-6 double to -2x+6y=-12

The system

is equivalent to

and furthermore equivalent to

where w = -2x+6y is some unknown single number.
Key term: single
The value of w cannot be equal to more than one number at the same time.
We get another contradiction like what happened with method 1.

------------------

Method 3

Let's solve each equation for y.

and

We arrive at equations of the form (slope-intercept form)
Each slope is m = 1/3
But the y intercepts are different (-2 and 1)
This tells us we have two parallel lines that never cross.
Parallel lines have equal slope, but different y intercepts.

We need the lines to cross somewhere to generate a solution.

Graph:

aka in green
aka in blue

Desmos and GeoGebra are two graphing options I recommend.

------------------

Method 4

Substitution

There are a number of ways to do substitution.
I'll go over one of them. Feel free to explore other methods.

Let's solve for x in the 1st equation.
-x+3y=-6
-x=-6-3y
x = 6+3y

Then we'll plug this into the 2nd equation to solve for y.
6y = 2x+6
6y = 2( x ) + 6
6y = 2(6+3y)+6
6y = 12+6y+6
6y-6y = 12+6y+6-6y
0y = 18+0y
0 = 18+0
0 = 18
We run into another contradiction.
This is further proof the system is inconsistent with no solutions

Answer by MathTherapy(10556)   (Show Source):
You can put this solution on YOUR website!

-x+3y=-6 6y=2x+6 - x + 3y = - 6 ------ eq (i) 6y = 2x + 6____2x - 6y = - 6 ---- eq (ii) SIMPLY multiply eq (i) by - 2 to get 2x - 6y = 12, and you'll see that the left sides of the altered equation and eq (ii) are the SAME. This means that the graphs of these linear equations are PARALLEL. Or, SIMPLY divide eq (ii) by common factor - 2 to get the same left side as eq (i). Again, the graphs of the 2 linear equations are PARALLEL. BTW, where's your question?