SOLUTION: Use substitution or elimination to solve each system of equations. Please explain. 7. 2x + 5y=16 5x - 2y= 11

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Question 120188: Use substitution or elimination to solve each system of equations. Please explain.

7. 2x + 5y=16
5x - 2y= 11
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

2%2Ax%2B5%2Ay=16
5%2Ax-2%2Ay=11

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=16-2%2AxSubtract 2%2Ax from both sides

y=%2816-2%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=16%2F5-%282%2F5%29%2Ax Now we've fully isolated y

Since y equals 16%2F5-%282%2F5%29%2Ax we can substitute the expression 16%2F5-%282%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


5%2Ax%2B-2%2Ahighlight%28%2816%2F5-%282%2F5%29%2Ax%29%29=11 Replace y with 16%2F5-%282%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

5%2Ax-2%2A%2816%2F5%29-2%28-2%2F5%29x=11 Distribute -2 to 16%2F5-%282%2F5%29%2Ax

5%2Ax-32%2F5%2B%284%2F5%29%2Ax=11 Multiply



5%2Ax-32%2F5%2B%284%2F5%29%2Ax=11 Reduce any fractions

5%2Ax%2B%284%2F5%29%2Ax=11%2B32%2F5Add 32%2F5 to both sides


5%2Ax%2B%284%2F5%29%2Ax=55%2F5%2B32%2F5 Make 11 into a fraction with a denominator of 5


5%2Ax%2B%284%2F5%29%2Ax=87%2F5 Combine the terms on the right side



%2825%2F5%29%2Ax%2B%284%2F5%29x=87%2F5 Make 5 into a fraction with a denominator of 5

%2829%2F5%29%2Ax=87%2F5 Now combine the terms on the left side.


cross%28%285%2F29%29%2829%2F5%29%29x=%2887%2F5%29%285%2F29%29 Multiply both sides by 5%2F29. This will cancel out 29%2F5 and isolate x

So when we multiply 87%2F5 and 5%2F29 (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

5%283%29-2%2Ay=11 Plug in x=3 into the 2nd equation

15-2%2Ay=11 Multiply

-2%2Ay=11-15Subtract 15 from both sides

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

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

y=-4%2F-2 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=3 and y=2

which can also look like

(3,2)

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

2%2Ax%2B5%2Ay=16
5%2Ax-2%2Ay=11

we get


graph of 2%2Ax%2B5%2Ay=16 (red) and 5%2Ax-2%2Ay=11 (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 (3,2). This verifies our answer.


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Check:

Plug in (3,2) into the system of equations


Let x=3 and y=2. Now plug those values into the equation 2%2Ax%2B5%2Ay=16

2%2A%283%29%2B5%2A%282%29=16 Plug in x=3 and y=2


6%2B10=16 Multiply


16=16 Add


16=16 Reduce. Since this equation is true the solution works.


So the solution (3,2) satisfies 2%2Ax%2B5%2Ay=16



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

5%2A%283%29-2%2A%282%29=11 Plug in x=3 and y=2


15-4=11 Multiply


11=11 Add


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


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


Since the solution (3,2) satisfies the system of equations


2%2Ax%2B5%2Ay=16
5%2Ax-2%2Ay=11


this verifies our answer.