SOLUTION: Solve by substitution or elimination and circle the method you have chosen. (2x + 3y = -11 (4x - 2y = 2

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Question 142156: Solve by substitution or elimination and circle the method you have chosen.
(2x + 3y = -11
(4x - 2y = 2
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
by substitution
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax%2B3%2Ay=-11
4%2Ax-2%2Ay=2

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=-11-2%2AxSubtract 2%2Ax from both sides

y=%28-11-2%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=-11%2F3-%282%2F3%29%2Ax Now we've fully isolated y

Since y equals -11%2F3-%282%2F3%29%2Ax we can substitute the expression -11%2F3-%282%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B-2%2Ahighlight%28%28-11%2F3-%282%2F3%29%2Ax%29%29=2 Replace y with -11%2F3-%282%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax-2%2A%28-11%2F3%29-2%28-2%2F3%29x=2 Distribute -2 to -11%2F3-%282%2F3%29%2Ax

4%2Ax%2B22%2F3%2B%284%2F3%29%2Ax=2 Multiply



4%2Ax%2B22%2F3%2B%284%2F3%29%2Ax=2 Reduce any fractions

4%2Ax%2B%284%2F3%29%2Ax=2-22%2F3 Subtract 22%2F3 from both sides


4%2Ax%2B%284%2F3%29%2Ax=6%2F3-22%2F3 Make 2 into a fraction with a denominator of 3


4%2Ax%2B%284%2F3%29%2Ax=-16%2F3 Combine the terms on the right side



%2812%2F3%29%2Ax%2B%284%2F3%29x=-16%2F3 Make 4 into a fraction with a denominator of 3

%2816%2F3%29%2Ax=-16%2F3 Now combine the terms on the left side.


cross%28%283%2F16%29%2816%2F3%29%29x=%28-16%2F3%29%283%2F16%29 Multiply both sides by 3%2F16. This will cancel out 16%2F3 and isolate x

So when we multiply -16%2F3 and 3%2F16 (and simplify) we get



x=-1 <---------------------------------One answer

Now that we know that x=-1, lets substitute that in for x to solve for y

4%28-1%29-2%2Ay=2 Plug in x=-1 into the 2nd equation

-4-2%2Ay=2 Multiply

-2%2Ay=2%2B4Add 4 to both sides

-2%2Ay=6 Combine the terms on the right side

cross%28%281%2F-2%29%28-2%29%29%2Ay=%286%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=6%2F-2 Multiply the terms on the right side


y=-3 Reduce


So this is the other answer


y=-3<---------------------------------Other answer


So our solution is

x=-1 and y=-3

which can also look like

(-1,-3)

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

2%2Ax%2B3%2Ay=-11
4%2Ax-2%2Ay=2

we get


graph of 2%2Ax%2B3%2Ay=-11 (red) and 4%2Ax-2%2Ay=2 (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 (-1,-3). This verifies our answer.


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Check:

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


Let x=-1 and y=-3. Now plug those values into the equation 2%2Ax%2B3%2Ay=-11

2%2A%28-1%29%2B3%2A%28-3%29=-11 Plug in x=-1 and y=-3


-2-9=-11 Multiply


-11=-11 Add


-11=-11 Reduce. Since this equation is true the solution works.


So the solution (-1,-3) satisfies 2%2Ax%2B3%2Ay=-11



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

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


-4%2B6=2 Multiply


2=2 Add


2=2 Reduce. Since this equation is true the solution works.


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


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


2%2Ax%2B3%2Ay=-11
4%2Ax-2%2Ay=2


this verifies our answer.