SOLUTION: Actually, I'm not sure what section this is in, but I really need help! Please, if you could help me, I need to solve the following equation: 4x+6y=3 2x+3y=-2

Algebra ->  Expressions-with-variables -> SOLUTION: Actually, I'm not sure what section this is in, but I really need help! Please, if you could help me, I need to solve the following equation: 4x+6y=3 2x+3y=-2      Log On


   

Question 117242: Actually, I'm not sure what section this is in, but I really need help! Please, if you could help me, I need to solve the following equation:
4x+6y=3
2x+3y=-2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

4%2Ax%2B6%2Ay=3
2%2Ax%2B3%2Ay=-2

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

6%2Ay=3-4%2AxSubtract 4%2Ax from both sides

y=%283-4%2Ax%29%2F6 Divide both sides by 6.


Which breaks down and reduces to



y=1%2F2-%282%2F3%29%2Ax Now we've fully isolated y

Since y equals 1%2F2-%282%2F3%29%2Ax we can substitute the expression 1%2F2-%282%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B3%2Ahighlight%28%281%2F2-%282%2F3%29%2Ax%29%29=-2 Replace y with 1%2F2-%282%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B3%2A%281%2F2%29%2B3%28-2%2F3%29x=-2 Distribute 3 to 1%2F2-%282%2F3%29%2Ax

2%2Ax%2B3%2F2-%286%2F3%29%2Ax=-2 Multiply



2%2Ax%2B3%2F2-2%2Ax=-2 Reduce any fractions

2%2Ax-2%2Ax=-2-3%2F2 Subtract 3%2F2 from both sides


2%2Ax-2%2Ax=-4%2F2-3%2F2 Make -2 into a fraction with a denominator of 2


2%2Ax-2%2Ax=-7%2F2 Combine the terms on the right side



0%2Ax=-7%2F2 Now combine the terms on the left side.
0%2F1=-7%2F2 Since this expression is not true, we have an inconsistency.


So there are no solutions. The simple reason is the 2 equations represent 2 parallel lines that will never intersect. Since no intersections occur, no solutions exist.


+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%283-4%2Ax%29%2F6%2C+%28-2-2%2Ax%29%2F3+%29+ graph of 4%2Ax%2B6%2Ay=3 (red) and 2%2Ax%2B3%2Ay=-2 (green) (hint: you may have to solve for y to graph these)


and we can see that the two equations are parallel and will never intersect. So this system is inconsistent