SOLUTION: please help me solve by substitution: 4x+3y=-3 x+y=-3

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Question 110098: please help me solve by substitution:
4x+3y=-3
x+y=-3

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

4%2Ax%2B3%2Ay=-3
1%2Ax%2B1%2Ay=-3

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

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


Which breaks down and reduces to



y=-1-%284%2F3%29%2Ax Now we've fully isolated y

Since y equals -1-%284%2F3%29%2Ax we can substitute the expression -1-%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%28-1-%284%2F3%29%2Ax%29%29=-3 Replace y with -1-%284%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%28-1%29%2B1%28-4%2F3%29x=-3 Distribute 1 to -1-%284%2F3%29%2Ax

1%2Ax-1-%284%2F3%29%2Ax=-3 Multiply



1%2Ax-1-%284%2F3%29%2Ax=-3 Reduce any fractions

1%2Ax-%284%2F3%29%2Ax=-3%2B1Add 1 to both sides


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



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

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


cross%28%283%2F-1%29%28-1%2F3%29%29x=%28-2%2F1%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 -2%2F1 and 3%2F-1 (and simplify) we get



x=6 <---------------------------------One answer

Now that we know that x=6, lets substitute that in for x to solve for y

1%286%29%2B1%2Ay=-3 Plug in x=6 into the 2nd equation

6%2B1%2Ay=-3 Multiply

1%2Ay=-3-6Subtract 6 from both sides

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

cross%28%281%2F1%29%281%29%29%2Ay=%28-9%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=-9%2F1 Multiply the terms on the right side


y=-9 Reduce


So this is the other answer


y=-9<---------------------------------Other answer


So our solution is

x=6 and y=-9

which can also look like

(6,-9)

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

4%2Ax%2B3%2Ay=-3
1%2Ax%2B1%2Ay=-3

we get


graph of 4%2Ax%2B3%2Ay=-3 (red) and 1%2Ax%2B1%2Ay=-3 (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 (6,-9). This verifies our answer.


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Check:

Plug in (6,-9) into the system of equations


Let x=6 and y=-9. Now plug those values into the equation 4%2Ax%2B3%2Ay=-3

4%2A%286%29%2B3%2A%28-9%29=-3 Plug in x=6 and y=-9


24-27=-3 Multiply


-3=-3 Add


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


So the solution (6,-9) satisfies 4%2Ax%2B3%2Ay=-3



Let x=6 and y=-9. Now plug those values into the equation 1%2Ax%2B1%2Ay=-3

1%2A%286%29%2B1%2A%28-9%29=-3 Plug in x=6 and y=-9


6-9=-3 Multiply


-3=-3 Add


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


So the solution (6,-9) satisfies 1%2Ax%2B1%2Ay=-3


Since the solution (6,-9) satisfies the system of equations


4%2Ax%2B3%2Ay=-3
1%2Ax%2B1%2Ay=-3


this verifies our answer.