SOLUTION: solve by substitution 1. 2x - y= -4 x+y = -5 2. 5x - 6y=21 x -2y=5

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Question 126980: solve by substitution
1. 2x - y= -4
x+y = -5

2. 5x - 6y=21
x -2y=5
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

solve by substitution
1.
2x+-+y=+-4
x%2By+=+-5

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


Lets start with the given system of linear equations

2%2Ax-1%2Ay=-4
1%2Ax%2B1%2Ay=-5

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=-4-2%2AxSubtract 2%2Ax from both sides

y=%28-4-2%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=4%2B2%2Ax Now we've fully isolated y

Since y equals 4%2B2%2Ax we can substitute the expression 4%2B2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%284%2B2%2Ax%29%29=-5 Replace y with 4%2B2%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%284%29%2B1%282%29x=-5 Distribute 1 to 4%2B2%2Ax

1%2Ax%2B4%2B2%2Ax=-5 Multiply



1%2Ax%2B4%2B2%2Ax=-5 Reduce any fractions

1%2Ax%2B2%2Ax=-5-4 Subtract 4 from both sides


1%2Ax%2B2%2Ax=-9 Combine the terms on the right side



3%2Ax=-9 Now combine the terms on the left side.


cross%28%281%2F3%29%283%2F1%29%29x=%28-9%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3%2F1 and isolate x

So when we multiply -9%2F1 and 1%2F3 (and simplify) we get



x=-3 <---------------------------------One answer

Now that we know that x=-3, lets substitute that in for x to solve for y

1%28-3%29%2B1%2Ay=-5 Plug in x=-3 into the 2nd equation

-3%2B1%2Ay=-5 Multiply

1%2Ay=-5%2B3Add 3 to both sides

1%2Ay=-2 Combine the terms on the right side

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

y=-2%2F1 Multiply the terms on the right side


y=-2 Reduce


So this is the other answer


y=-2<---------------------------------Other answer


So our solution is

x=-3 and y=-2

which can also look like

(-3,-2)

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

2%2Ax-1%2Ay=-4
1%2Ax%2B1%2Ay=-5

we get


graph of 2%2Ax-1%2Ay=-4 (red) and 1%2Ax%2B1%2Ay=-5 (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 (-3,-2). This verifies our answer.


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

Plug in (-3,-2) into the system of equations


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

2%2A%28-3%29-1%2A%28-2%29=-4 Plug in x=-3 and y=-2


-6%2B2=-4 Multiply


-4=-4 Add


-4=-4 Reduce. Since this equation is true the solution works.


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



Let x=-3 and y=-2. Now plug those values into the equation 1%2Ax%2B1%2Ay=-5

1%2A%28-3%29%2B1%2A%28-2%29=-5 Plug in x=-3 and y=-2


-3-2=-5 Multiply


-5=-5 Add


-5=-5 Reduce. Since this equation is true the solution works.


So the solution (-3,-2) satisfies 1%2Ax%2B1%2Ay=-5


Since the solution (-3,-2) satisfies the system of equations


2%2Ax-1%2Ay=-4
1%2Ax%2B1%2Ay=-5


this verifies our answer.




2.
5x+-6y=21
x+-2y=5
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

5%2Ax-6%2Ay=21
1%2Ax-2%2Ay=5

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=21-5%2AxSubtract 5%2Ax from both sides

y=%2821-5%2Ax%29%2F-6 Divide both sides by -6.


Which breaks down and reduces to



y=-7%2F2%2B%285%2F6%29%2Ax Now we've fully isolated y

Since y equals -7%2F2%2B%285%2F6%29%2Ax we can substitute the expression -7%2F2%2B%285%2F6%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-2%2Ahighlight%28%28-7%2F2%2B%285%2F6%29%2Ax%29%29=5 Replace y with -7%2F2%2B%285%2F6%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-2%2A%28-7%2F2%29-2%285%2F6%29x=5 Distribute -2 to -7%2F2%2B%285%2F6%29%2Ax

1%2Ax%2B14%2F2-%2810%2F6%29%2Ax=5 Multiply



1%2Ax%2B7-%285%2F3%29%2Ax=5 Reduce any fractions

1%2Ax-%285%2F3%29%2Ax=5-7 Subtract 7 from both sides


1%2Ax-%285%2F3%29%2Ax=-2 Combine the terms on the right side



%283%2F3%29%2Ax-%285%2F3%29x=-2 Make 1 into a fraction with a denominator of 3

%28-2%2F3%29%2Ax=-2 Now combine the terms on the left side.


cross%28%283%2F-2%29%28-2%2F3%29%29x=%28-2%2F1%29%283%2F-2%29 Multiply both sides by 3%2F-2. This will cancel out -2%2F3 and isolate x

So when we multiply -2%2F1 and 3%2F-2 (and simplify) we get



x=3 <---------------------------------One answer

Now that we know that x=3, lets substitute that in for x to solve for y

1%283%29-2%2Ay=5 Plug in x=3 into the 2nd equation

3-2%2Ay=5 Multiply

-2%2Ay=5-3Subtract 3 from both sides

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

cross%28%281%2F-2%29%28-2%29%29%2Ay=%282%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=2%2F-2 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=3 and y=-1

which can also look like

(3,-1)

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

5%2Ax-6%2Ay=21
1%2Ax-2%2Ay=5

we get


graph of 5%2Ax-6%2Ay=21 (red) and 1%2Ax-2%2Ay=5 (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 (3,-1). This verifies our answer.


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

Plug in (3,-1) into the system of equations


Let x=3 and y=-1. Now plug those values into the equation 5%2Ax-6%2Ay=21

5%2A%283%29-6%2A%28-1%29=21 Plug in x=3 and y=-1


15%2B6=21 Multiply


21=21 Add


21=21 Reduce. Since this equation is true the solution works.


So the solution (3,-1) satisfies 5%2Ax-6%2Ay=21



Let x=3 and y=-1. Now plug those values into the equation 1%2Ax-2%2Ay=5

1%2A%283%29-2%2A%28-1%29=5 Plug in x=3 and y=-1


3%2B2=5 Multiply


5=5 Add


5=5 Reduce. Since this equation is true the solution works.


So the solution (3,-1) satisfies 1%2Ax-2%2Ay=5


Since the solution (3,-1) satisfies the system of equations


5%2Ax-6%2Ay=21
1%2Ax-2%2Ay=5


this verifies our answer.