SOLUTION: solve by substitution 1. 4x + 5y= 6 y=2x - 10 2. 4x-3y=0 y=x+1

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Question 126964: solve by substitution
1. 4x + 5y= 6
y=2x - 10
2. 4x-3y=0
y=x+1
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

1.
4x+%2B+5y=+6
y=2x+-10.....write in standard form
4x+%2B+5y=+6
-2x+%2B+y=+-10

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


Lets start with the given system of linear equations

4%2Ax%2B5%2Ay=6
-2%2Ax%2B1%2Ay=-10

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

5%2Ay=6-4%2AxSubtract 4%2Ax from both sides

y=%286-4%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=6%2F5-%284%2F5%29%2Ax Now we've fully isolated y

Since y equals 6%2F5-%284%2F5%29%2Ax we can substitute the expression 6%2F5-%284%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-2%2Ax%2B1%2Ahighlight%28%286%2F5-%284%2F5%29%2Ax%29%29=-10 Replace y with 6%2F5-%284%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

-2%2Ax%2B1%2A%286%2F5%29%2B1%28-4%2F5%29x=-10 Distribute 1 to 6%2F5-%284%2F5%29%2Ax

-2%2Ax%2B6%2F5-%284%2F5%29%2Ax=-10 Multiply



-2%2Ax%2B6%2F5-%284%2F5%29%2Ax=-10 Reduce any fractions

-2%2Ax-%284%2F5%29%2Ax=-10-6%2F5 Subtract 6%2F5 from both sides


-2%2Ax-%284%2F5%29%2Ax=-50%2F5-6%2F5 Make -10 into a fraction with a denominator of 5


-2%2Ax-%284%2F5%29%2Ax=-56%2F5 Combine the terms on the right side



%28-10%2F5%29%2Ax-%284%2F5%29x=-56%2F5 Make -2 into a fraction with a denominator of 5

%28-14%2F5%29%2Ax=-56%2F5 Now combine the terms on the left side.


cross%28%285%2F-14%29%28-14%2F5%29%29x=%28-56%2F5%29%285%2F-14%29 Multiply both sides by 5%2F-14. This will cancel out -14%2F5 and isolate x

So when we multiply -56%2F5 and 5%2F-14 (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%2B1%2Ay=-10 Plug in x=4 into the 2nd equation

-8%2B1%2Ay=-10 Multiply

1%2Ay=-10%2B8Add 8 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=4 and y=-2

which can also look like

(4,-2)

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

4%2Ax%2B5%2Ay=6
-2%2Ax%2B1%2Ay=-10

we get


graph of 4%2Ax%2B5%2Ay=6 (red) and -2%2Ax%2B1%2Ay=-10 (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,-2). This verifies our answer.


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

Plug in (4,-2) into the system of equations


Let x=4 and y=-2. Now plug those values into the equation 4%2Ax%2B5%2Ay=6

4%2A%284%29%2B5%2A%28-2%29=6 Plug in x=4 and y=-2


16-10=6 Multiply


6=6 Add


6=6 Reduce. Since this equation is true the solution works.


So the solution (4,-2) satisfies 4%2Ax%2B5%2Ay=6



Let x=4 and y=-2. Now plug those values into the equation -2%2Ax%2B1%2Ay=-10

-2%2A%284%29%2B1%2A%28-2%29=-10 Plug in x=4 and y=-2


-8-2=-10 Multiply


-10=-10 Add


-10=-10 Reduce. Since this equation is true the solution works.


So the solution (4,-2) satisfies -2%2Ax%2B1%2Ay=-10


Since the solution (4,-2) satisfies the system of equations


4%2Ax%2B5%2Ay=6
-2%2Ax%2B1%2Ay=-10


this verifies our answer.




2.
4x+-+3y=+0
y=x+%2B1.....write in standard form
4x+-+3y=+0
-x+%2B+y=+1
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

4%2Ax-3%2Ay=0
-1%2Ax%2B1%2Ay=1

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

-3%2Ay=0-4%2AxSubtract 4%2Ax from both sides

y=%280-4%2Ax%29%2F-3 Divide both sides by -3.


Which breaks down and reduces to



y=0%2B%284%2F3%29%2Ax Now we've fully isolated y

Since y equals 0%2B%284%2F3%29%2Ax we can substitute the expression 0%2B%284%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-1%2Ax%2B1%2Ahighlight%28%280%2B%284%2F3%29%2Ax%29%29=1 Replace y with 0%2B%284%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

-1%2Ax%2B1%2A%280%29%2B1%284%2F3%29x=1 Distribute 1 to 0%2B%284%2F3%29%2Ax

-1%2Ax%2B0%2B%284%2F3%29%2Ax=1 Multiply



-1%2Ax%2B0%2B%284%2F3%29%2Ax=1 Reduce any fractions

-1%2Ax%2B%284%2F3%29%2Ax=1%2B0Add 0 to both sides


-1%2Ax%2B%284%2F3%29%2Ax=1 Combine the terms on the right side



%28-3%2F3%29%2Ax%2B%284%2F3%29x=1 Make -1 into a fraction with a denominator of 3

%281%2F3%29%2Ax=1 Now combine the terms on the left side.


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

So when we multiply 1%2F1 and 3%2F1 (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%2B1%2Ay=1 Plug in x=3 into the 2nd equation

-3%2B1%2Ay=1 Multiply

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

1%2Ay=4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%284%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=4%2F1 Multiply the terms on the right side


y=4 Reduce


So this is the other answer


y=4<---------------------------------Other answer


So our solution is

x=3 and y=4

which can also look like

(3,4)

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

4%2Ax-3%2Ay=0
-1%2Ax%2B1%2Ay=1

we get


graph of 4%2Ax-3%2Ay=0 (red) and -1%2Ax%2B1%2Ay=1 (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 (3,4). This verifies our answer.


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

Plug in (3,4) into the system of equations


Let x=3 and y=4. Now plug those values into the equation 4%2Ax-3%2Ay=0

4%2A%283%29-3%2A%284%29=0 Plug in x=3 and y=4


12-12=0 Multiply


0=0 Add


0=0 Reduce. Since this equation is true the solution works.


So the solution (3,4) satisfies 4%2Ax-3%2Ay=0



Let x=3 and y=4. Now plug those values into the equation -1%2Ax%2B1%2Ay=1

-1%2A%283%29%2B1%2A%284%29=1 Plug in x=3 and y=4


-3%2B4=1 Multiply


1=1 Add


1=1 Reduce. Since this equation is true the solution works.


So the solution (3,4) satisfies -1%2Ax%2B1%2Ay=1


Since the solution (3,4) satisfies the system of equations


4%2Ax-3%2Ay=0
-1%2Ax%2B1%2Ay=1


this verifies our answer.