SOLUTION: Solve each system using substitution.write no solution or infinitely many solutions where appropriate.(1)x=-2y+1. x=y-5 (2) y=x-7.2x+y=8 (3)3x+y=10.y=-3x+4 (4)4x+2y=8.y=-2x+4 (5)5x

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Question 1010610: Solve each system using substitution.write no solution or infinitely many solutions where appropriate.(1)x=-2y+1. x=y-5 (2) y=x-7.2x+y=8 (3)3x+y=10.y=-3x+4 (4)4x+2y=8.y=-2x+4 (5)5x-3y=-4.x+y=-4 how to solve this?
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!

(1)


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


Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations




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

Subtract from both sides

Divide both sides by 2.


Which breaks down and reduces to



Now we've fully isolated y

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


Replace y with . Since this eliminates y, we can now solve for x.

Distribute -1 to

Multiply



Reduce any fractions

Add to both sides


Make -5 into a fraction with a denominator of 2


Combine the terms on the right side



Make 1 into a fraction with a denominator of 2

Now combine the terms on the left side.


Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get



<---------------------------------One answer

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

Plug in into the 2nd equation

Multiply

Add to both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out -1 on the left side.

Multiply the terms on the right side


Reduce


So this is the other answer


<---------------------------------Other answer


So our solution is

and

which can also look like

(,)

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




we get


graph of (red) and (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 (,). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (,) into the system of equations


Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies



Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies


Since the solution (,) satisfies the system of equations






this verifies our answer.



(2)


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


Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations




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

Add to both sides

Divide both sides by 1.


Which breaks down and reduces to



Now we've fully isolated y

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


Replace y with . Since this eliminates y, we can now solve for x.

Distribute 1 to

Multiply



Reduce any fractions

Add to both sides


Combine the terms on the right side



Now combine the terms on the left side.


Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get



<---------------------------------One answer

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

Plug in into the 2nd equation

Multiply

Subtract from both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out 1 on the left side.

Multiply the terms on the right side


Reduce


So this is the other answer


<---------------------------------Other answer


So our solution is

and

which can also look like

(,)

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




we get


graph of (red) and (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 (,). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (,) into the system of equations


Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies



Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies


Since the solution (,) satisfies the system of equations






this verifies our answer.





(3)


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


Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations




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

Subtract from both sides

Divide both sides by 1.


Which breaks down and reduces to



Now we've fully isolated y

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


Replace y with . Since this eliminates y, we can now solve for x.

Distribute 1 to

Multiply



Reduce any fractions

Subtract from both sides


Combine the terms on the right side



Now combine the terms on the left side.
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 of (red) and (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


(4)


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


Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations




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

Subtract from both sides

Divide both sides by 2.


Which breaks down and reduces to



Now we've fully isolated y

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


Replace y with . Since this eliminates y, we can now solve for x.

Distribute 1 to

Multiply



Reduce any fractions

Subtract from both sides


Combine the terms on the right side



Now combine the terms on the left side.
Since this expression is true for any x, we have an identity.


So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points.

If we graph and we get

graph of


graph of (hint: you may have to solve for y to graph these)

we can see that these two lines are the same. So this system is dependent


(5)


---------------------------
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations




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

Subtract from both sides

Divide both sides by -3.


Which breaks down and reduces to



Now we've fully isolated y

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


Replace y with . Since this eliminates y, we can now solve for x.

Distribute 1 to

Multiply



Reduce any fractions

Subtract from both sides


Make -4 into a fraction with a denominator of 3


Combine the terms on the right side



Make 1 into a fraction with a denominator of 3

Now combine the terms on the left side.


Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get



<---------------------------------One answer

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

Plug in into the 2nd equation

Multiply

Add to both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out 1 on the left side.

Multiply the terms on the right side


Reduce


So this is the other answer


<---------------------------------Other answer


So our solution is

and

which can also look like

(,)

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




we get


graph of (red) and (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 (,). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (,) into the system of equations


Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies



Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies


Since the solution (,) satisfies the system of equations






this verifies our answer.
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