SOLUTION: Solve the system; 3x+4y=23 2x-5y=-23

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Question 101103: Solve the system;
3x+4y=23
2x-5y=-23
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

3%2Ax%2B4%2Ay=23
2%2Ax-5%2Ay=-23

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

4%2Ay=23-3%2AxSubtract 3%2Ax from both sides

y=%2823-3%2Ax%29%2F4 Divide both sides by 4.


Which breaks down and reduces to



y=23%2F4-%283%2F4%29%2Ax Now we've fully isolated y

Since y equals 23%2F4-%283%2F4%29%2Ax we can substitute the expression 23%2F4-%283%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B-5%2Ahighlight%28%2823%2F4-%283%2F4%29%2Ax%29%29=-23 Replace y with 23%2F4-%283%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax-5%2A%2823%2F4%29-5%28-3%2F4%29x=-23 Distribute -5 to 23%2F4-%283%2F4%29%2Ax

2%2Ax-115%2F4%2B%2815%2F4%29%2Ax=-23 Multiply



2%2Ax-115%2F4%2B%2815%2F4%29%2Ax=-23 Reduce any fractions

2%2Ax%2B%2815%2F4%29%2Ax=-23%2B115%2F4Add 115%2F4 to both sides


2%2Ax%2B%2815%2F4%29%2Ax=-92%2F4%2B115%2F4 Make -23 into a fraction with a denominator of 4


2%2Ax%2B%2815%2F4%29%2Ax=23%2F4 Combine the terms on the right side



%288%2F4%29%2Ax%2B%2815%2F4%29x=23%2F4 Make 2 into a fraction with a denominator of 4

%2823%2F4%29%2Ax=23%2F4 Now combine the terms on the left side.


cross%28%284%2F23%29%2823%2F4%29%29x=%2823%2F4%29%284%2F23%29 Multiply both sides by 4%2F23. This will cancel out 23%2F4 and isolate x

So when we multiply 23%2F4 and 4%2F23 (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

2%281%29-5%2Ay=-23 Plug in x=1 into the 2nd equation

2-5%2Ay=-23 Multiply

-5%2Ay=-23-2Subtract 2 from both sides

-5%2Ay=-25 Combine the terms on the right side

cross%28%281%2F-5%29%28-5%29%29%2Ay=%28-25%2F1%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5 on the left side.

y=-25%2F-5 Multiply the terms on the right side


y=5 Reduce


So this is the other answer


y=5<---------------------------------Other answer


So our solution is

x=1 and y=5

which can also look like

(1,5)

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

3%2Ax%2B4%2Ay=23
2%2Ax-5%2Ay=-23

we get


graph of 3%2Ax%2B4%2Ay=23 (red) and 2%2Ax-5%2Ay=-23 (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 (1,5). This verifies our answer.


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Check:

Plug in (1,5) into the system of equations


Let x=1 and y=5. Now plug those values into the equation 3%2Ax%2B4%2Ay=23

3%2A%281%29%2B4%2A%285%29=23 Plug in x=1 and y=5


3%2B20=23 Multiply


23=23 Add


23=23 Reduce. Since this equation is true the solution works.


So the solution (1,5) satisfies 3%2Ax%2B4%2Ay=23



Let x=1 and y=5. Now plug those values into the equation 2%2Ax-5%2Ay=-23

2%2A%281%29-5%2A%285%29=-23 Plug in x=1 and y=5


2-25=-23 Multiply


-23=-23 Add


-23=-23 Reduce. Since this equation is true the solution works.


So the solution (1,5) satisfies 2%2Ax-5%2Ay=-23


Since the solution (1,5) satisfies the system of equations


3%2Ax%2B4%2Ay=23
2%2Ax-5%2Ay=-23


this verifies our answer.