SOLUTION: solve each systems of equations algebraically 36. x+5y=11 3x+2y=46

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Question 120663: solve each systems of equations algebraically

36. x+5y=11
3x+2y=46
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

#36

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


Lets start with the given system of linear equations

1%2Ax%2B5%2Ay=11
3%2Ax%2B2%2Ay=46

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=11-1%2AxSubtract 1%2Ax from both sides

y=%2811-1%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=11%2F5-%281%2F5%29%2Ax Now we've fully isolated y

Since y equals 11%2F5-%281%2F5%29%2Ax we can substitute the expression 11%2F5-%281%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B2%2Ahighlight%28%2811%2F5-%281%2F5%29%2Ax%29%29=46 Replace y with 11%2F5-%281%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B2%2A%2811%2F5%29%2B2%28-1%2F5%29x=46 Distribute 2 to 11%2F5-%281%2F5%29%2Ax

3%2Ax%2B22%2F5-%282%2F5%29%2Ax=46 Multiply



3%2Ax%2B22%2F5-%282%2F5%29%2Ax=46 Reduce any fractions

3%2Ax-%282%2F5%29%2Ax=46-22%2F5 Subtract 22%2F5 from both sides


3%2Ax-%282%2F5%29%2Ax=230%2F5-22%2F5 Make 46 into a fraction with a denominator of 5


3%2Ax-%282%2F5%29%2Ax=208%2F5 Combine the terms on the right side



%2815%2F5%29%2Ax-%282%2F5%29x=208%2F5 Make 3 into a fraction with a denominator of 5

%2813%2F5%29%2Ax=208%2F5 Now combine the terms on the left side.


cross%28%285%2F13%29%2813%2F5%29%29x=%28208%2F5%29%285%2F13%29 Multiply both sides by 5%2F13. This will cancel out 13%2F5 and isolate x

So when we multiply 208%2F5 and 5%2F13 (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

3%2816%29%2B2%2Ay=46 Plug in x=16 into the 2nd equation

48%2B2%2Ay=46 Multiply

2%2Ay=46-48Subtract 48 from both sides

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

cross%28%281%2F2%29%282%29%29%2Ay=%28-2%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=-2%2F2 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=16 and y=-1

which can also look like

(16,-1)

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

1%2Ax%2B5%2Ay=11
3%2Ax%2B2%2Ay=46

we get


graph of 1%2Ax%2B5%2Ay=11 (red) and 3%2Ax%2B2%2Ay=46 (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,-1). This verifies our answer.


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Check:

Plug in (16,-1) into the system of equations


Let x=16 and y=-1. Now plug those values into the equation 1%2Ax%2B5%2Ay=11

1%2A%2816%29%2B5%2A%28-1%29=11 Plug in x=16 and y=-1


16-5=11 Multiply


11=11 Add


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


So the solution (16,-1) satisfies 1%2Ax%2B5%2Ay=11



Let x=16 and y=-1. Now plug those values into the equation 3%2Ax%2B2%2Ay=46

3%2A%2816%29%2B2%2A%28-1%29=46 Plug in x=16 and y=-1


48-2=46 Multiply


46=46 Add


46=46 Reduce. Since this equation is true the solution works.


So the solution (16,-1) satisfies 3%2Ax%2B2%2Ay=46


Since the solution (16,-1) satisfies the system of equations


1%2Ax%2B5%2Ay=11
3%2Ax%2B2%2Ay=46


this verifies our answer.