SOLUTION: Please help me solve using substitution 2x - 5y = 11 -2x + 3y = -9

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Question 1010335: Please help me solve using substitution
2x - 5y = 11
-2x + 3y = -9
Answer by MathLover1(20850) 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

2%2Ax-5%2Ay=11
-2%2Ax%2B3%2Ay=-9

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

y=%2811-2%2Ax%29%2F-5 Divide both sides by -5.


Which breaks down and reduces to



y=-11%2F5%2B%282%2F5%29%2Ax Now we've fully isolated y

Since y equals -11%2F5%2B%282%2F5%29%2Ax we can substitute the expression -11%2F5%2B%282%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-2%2Ax%2B3%2Ahighlight%28%28-11%2F5%2B%282%2F5%29%2Ax%29%29=-9 Replace y with -11%2F5%2B%282%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

-2%2Ax%2B3%2A%28-11%2F5%29%2B3%282%2F5%29x=-9 Distribute 3 to -11%2F5%2B%282%2F5%29%2Ax

-2%2Ax-33%2F5%2B%286%2F5%29%2Ax=-9 Multiply



-2%2Ax-33%2F5%2B%286%2F5%29%2Ax=-9 Reduce any fractions

-2%2Ax%2B%286%2F5%29%2Ax=-9%2B33%2F5Add 33%2F5 to both sides


-2%2Ax%2B%286%2F5%29%2Ax=-45%2F5%2B33%2F5 Make -9 into a fraction with a denominator of 5


-2%2Ax%2B%286%2F5%29%2Ax=-12%2F5 Combine the terms on the right side



%28-10%2F5%29%2Ax%2B%286%2F5%29x=-12%2F5 Make -2 into a fraction with a denominator of 5

%28-4%2F5%29%2Ax=-12%2F5 Now combine the terms on the left side.


cross%28%285%2F-4%29%28-4%2F5%29%29x=%28-12%2F5%29%285%2F-4%29 Multiply both sides by 5%2F-4. This will cancel out -4%2F5 and isolate x

So when we multiply -12%2F5 and 5%2F-4 (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

-2%283%29%2B3%2Ay=-9 Plug in x=3 into the 2nd equation

-6%2B3%2Ay=-9 Multiply

3%2Ay=-9%2B6Add 6 to both sides

3%2Ay=-3 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ay=%28-3%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.

y=-3%2F3 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)

2%2Ax-5%2Ay=11
-2%2Ax%2B3%2Ay=-9

we get


graph of 2%2Ax-5%2Ay=11 (red) and -2%2Ax%2B3%2Ay=-9 (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.


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Check:

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


Let x=3 and y=-1. Now plug those values into the equation 2%2Ax-5%2Ay=11

2%2A%283%29-5%2A%28-1%29=11 Plug in x=3 and y=-1


6%2B5=11 Multiply


11=11 Add


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


So the solution (3,-1) satisfies 2%2Ax-5%2Ay=11



Let x=3 and y=-1. Now plug those values into the equation -2%2Ax%2B3%2Ay=-9

-2%2A%283%29%2B3%2A%28-1%29=-9 Plug in x=3 and y=-1


-6-3=-9 Multiply


-9=-9 Add


-9=-9 Reduce. Since this equation is true the solution works.


So the solution (3,-1) satisfies -2%2Ax%2B3%2Ay=-9


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


2%2Ax-5%2Ay=11
-2%2Ax%2B3%2Ay=-9


this verifies our answer.