SOLUTION: I need help on steps for solving a system of equations in substitution,elimination,and graphing method for problem: X+3y=-4 x+4=0 Thank you, This is how I tried to work it

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Question 922387: I need help on steps for solving a system of equations in substitution,elimination,and graphing method for problem:
X+3y=-4 x+4=0
Thank you,
This is how I tried to work it
Y=-x+0
X+3(-x+0)=-4
X+-3x+0=-4
-2x+0=-4
-2x\-2=-4\-2
X=2
(2,2)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
x%2B3y=-4
x%2B4=0+
_______________
1. by substitution

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


Lets start with the given system of linear equations

1%2Ax%2B3%2Ay=-4
1%2Ax%2B4%2Ay=0

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=-4-1%2AxSubtract 1%2Ax from both sides

y=%28-4-1%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=-4%2F3-%281%2F3%29%2Ax Now we've fully isolated y

Since y equals -4%2F3-%281%2F3%29%2Ax we can substitute the expression -4%2F3-%281%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B4%2Ahighlight%28%28-4%2F3-%281%2F3%29%2Ax%29%29=0 Replace y with -4%2F3-%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B4%2A%28-4%2F3%29%2B4%28-1%2F3%29x=0 Distribute 4 to -4%2F3-%281%2F3%29%2Ax

1%2Ax-16%2F3-%284%2F3%29%2Ax=0 Multiply



1%2Ax-16%2F3-%284%2F3%29%2Ax=0 Reduce any fractions

1%2Ax-%284%2F3%29%2Ax=0%2B16%2F3Add 16%2F3 to both sides


1%2Ax-%284%2F3%29%2Ax=0%2F3%2B16%2F3 Make 0 into a fraction with a denominator of 3


1%2Ax-%284%2F3%29%2Ax=16%2F3 Combine the terms on the right side



%283%2F3%29%2Ax-%284%2F3%29x=16%2F3 Make 1 into a fraction with a denominator of 3

%28-1%2F3%29%2Ax=16%2F3 Now combine the terms on the left side.


cross%28%283%2F-1%29%28-1%2F3%29%29x=%2816%2F3%29%283%2F-1%29 Multiply both sides by 3%2F-1. This will cancel out -1%2F3 and isolate x

So when we multiply 16%2F3 and 3%2F-1 (and simplify) we get



x=-16 <---------------------------------One answer

Now that we know that x=-16, lets substitute that in for x to solve for y

1%28-16%29%2B4%2Ay=0 Plug in x=-16 into the 2nd equation

-16%2B4%2Ay=0 Multiply

4%2Ay=0%2B16Add 16 to both sides

4%2Ay=16 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ay=%2816%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.

y=16%2F4 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=-16 and y=4

which can also look like

(-16,4)

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

1%2Ax%2B3%2Ay=-4
1%2Ax%2B4%2Ay=0

we get


graph of 1%2Ax%2B3%2Ay=-4 (red) and 1%2Ax%2B4%2Ay=0 (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 (-16,4). This verifies our answer.


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

Plug in (-16,4) into the system of equations


Let x=-16 and y=4. Now plug those values into the equation 1%2Ax%2B3%2Ay=-4

1%2A%28-16%29%2B3%2A%284%29=-4 Plug in x=-16 and y=4


-16%2B12=-4 Multiply


-4=-4 Add


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


So the solution (-16,4) satisfies 1%2Ax%2B3%2Ay=-4



Let x=-16 and y=4. Now plug those values into the equation 1%2Ax%2B4%2Ay=0

1%2A%28-16%29%2B4%2A%284%29=0 Plug in x=-16 and y=4


-16%2B16=0 Multiply


0=0 Add


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


So the solution (-16,4) satisfies 1%2Ax%2B4%2Ay=0


Since the solution (-16,4) satisfies the system of equations


1%2Ax%2B3%2Ay=-4
1%2Ax%2B4%2Ay=0


this verifies our answer.





2.by elimination
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B3%2Ay=-4
1%2Ax%2B4%2Ay=0

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 1 is 1, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -1 like this:

1%2A%281%2Ax%2B3%2Ay%29=%28-4%29%2A1 Multiply the top equation (both sides) by 1
-1%2A%281%2Ax%2B4%2Ay%29=%280%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
1%2Ax%2B3%2Ay=-4
-1%2Ax-4%2Ay=0

Notice how 1 and -1 add to zero (ie 1%2B-1=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%281%2Ax-1%2Ax%29%2B%283%2Ay-4%2Ay%29=-4%2B0

%281-1%29%2Ax%2B%283-4%29y=-4%2B0

cross%281%2B-1%29%2Ax%2B%283-4%29%2Ay=-4%2B0 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-1%2Ay=-4

y=-4%2F-1 Divide both sides by -1 to solve for y



y=4 Reduce


Now plug this answer into the top equation 1%2Ax%2B3%2Ay=-4 to solve for x

1%2Ax%2B3%284%29=-4 Plug in y=4


1%2Ax%2B12=-4 Multiply



1%2Ax=-4-12 Subtract 12 from both sides

1%2Ax=-16 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%28-16%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=-16 Multiply the terms on the right side


So our answer is

x=-16, y=4

which also looks like

(-16, 4)

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

1%2Ax%2B3%2Ay=-4
1%2Ax%2B4%2Ay=0

we get



graph of 1%2Ax%2B3%2Ay=-4 (red) 1%2Ax%2B4%2Ay=0 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (-16,4). This verifies our answer.


3.by graphing
Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


1x%2B3y=-4

1x%2B4y=0





In order to graph these equations, we need to solve for y for each equation.




So let's solve for y on the first equation


1x%2B3y=-4 Start with the given equation



3y=-4-x Subtract +x from both sides



3y=-x-4 Rearrange the equation



y=%28-x-4%29%2F%283%29 Divide both sides by 3



y=%28-1%2F3%29x%2B%28-4%29%2F%283%29 Break up the fraction



y=%28-1%2F3%29x-4%2F3 Reduce



Now lets graph y=%28-1%2F3%29x-4%2F3 (note: if you need help with graphing, check out this solver)



+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28-1%2F3%29x-4%2F3%29+ Graph of y=%28-1%2F3%29x-4%2F3




So let's solve for y on the second equation


1x%2B4y=0 Start with the given equation



4y=0-x Subtract +x from both sides



4y=-x%2B0 Rearrange the equation



y=%28-x%2B0%29%2F%284%29 Divide both sides by 4



y=%28-1%2F4%29x%2B%280%29%2F%284%29 Break up the fraction



y=%28-1%2F4%29x%2B0 Reduce





Now lets add the graph of y=%28-1%2F4%29x%2B0 to our first plot to get:


Graph of y=%28-1%2F3%29x-4%2F3(red) and y=%28-1%2F4%29x%2B0(green)


From the graph, we can see that the two lines intersect at the point (-16,4) (note: you might have to adjust the window to see the intersection)