SOLUTION: Solve the system of equation. If the system is consistent, state the coordinates of the point of intersection. If not consistent, state whether it is inconsistent of dependent.

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Solve the system of equation. If the system is consistent, state the coordinates of the point of intersection. If not consistent, state whether it is inconsistent of dependent.       Log On


   

Question 112307This question is from textbook
: Solve the system of equation. If the system is consistent, state the coordinates of the point of intersection. If not consistent, state whether it is inconsistent of dependent.
x-y=3
2x+3y=11 This question is from textbook

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

1%2Ax-1%2Ay=3
2%2Ax%2B3%2Ay=11

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

-1%2Ay=3-1%2AxSubtract 1%2Ax from both sides

y=%283-1%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-3%2B1%2Ax Now we've fully isolated y

Since y equals -3%2B1%2Ax we can substitute the expression -3%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B3%2Ahighlight%28%28-3%2B1%2Ax%29%29=11 Replace y with -3%2B1%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B3%2A%28-3%29%2B3%281%29x=11 Distribute 3 to -3%2B1%2Ax

2%2Ax-9%2B3%2Ax=11 Multiply



2%2Ax-9%2B3%2Ax=11 Reduce any fractions

2%2Ax%2B3%2Ax=11%2B9Add 9 to both sides


2%2Ax%2B3%2Ax=20 Combine the terms on the right side



5%2Ax=20 Now combine the terms on the left side.


cross%28%281%2F5%29%285%2F1%29%29x=%2820%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5%2F1 and isolate x

So when we multiply 20%2F1 and 1%2F5 (and simplify) we get



x=4 <---------------------------------One answer

Now that we know that x=4, lets substitute that in for x to solve for y

2%284%29%2B3%2Ay=11 Plug in x=4 into the 2nd equation

8%2B3%2Ay=11 Multiply

3%2Ay=11-8Subtract 8 from both sides

3%2Ay=3 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ay=%283%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.

y=3%2F3 Multiply the terms on the right side


y=1 Reduce


So this is the other answer


y=1<---------------------------------Other answer


So our solution is

x=4 and y=1

which can also look like

(4,1)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax-1%2Ay=3
2%2Ax%2B3%2Ay=11

we get


graph of 1%2Ax-1%2Ay=3 (red) and 2%2Ax%2B3%2Ay=11 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (4,1). This verifies our answer.


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Check:

Plug in (4,1) into the system of equations


Let x=4 and y=1. Now plug those values into the equation 1%2Ax-1%2Ay=3

1%2A%284%29-1%2A%281%29=3 Plug in x=4 and y=1


4-1=3 Multiply


3=3 Add


3=3 Reduce. Since this equation is true the solution works.


So the solution (4,1) satisfies 1%2Ax-1%2Ay=3



Let x=4 and y=1. Now plug those values into the equation 2%2Ax%2B3%2Ay=11

2%2A%284%29%2B3%2A%281%29=11 Plug in x=4 and y=1


8%2B3=11 Multiply


11=11 Add


11=11 Reduce. Since this equation is true the solution works.


So the solution (4,1) satisfies 2%2Ax%2B3%2Ay=11


Since the solution (4,1) satisfies the system of equations


1%2Ax-1%2Ay=3
2%2Ax%2B3%2Ay=11


this verifies our answer.