SOLUTION: solve the system by substitution. x - 5y = 2 2x + 3y = 17

Algebra ->  Graphs -> SOLUTION: solve the system by substitution. x - 5y = 2 2x + 3y = 17      Log On


   

Question 110529: solve the system by substitution.
x - 5y = 2
2x + 3y = 17
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

1%2Ax-5%2Ay=2
2%2Ax%2B3%2Ay=17

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

y=%282-1%2Ax%29%2F-5 Divide both sides by -5.


Which breaks down and reduces to



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

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


2%2Ax%2B3%2Ahighlight%28%28-2%2F5%2B%281%2F5%29%2Ax%29%29=17 Replace y with -2%2F5%2B%281%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B3%2A%28-2%2F5%29%2B3%281%2F5%29x=17 Distribute 3 to -2%2F5%2B%281%2F5%29%2Ax

2%2Ax-6%2F5%2B%283%2F5%29%2Ax=17 Multiply



2%2Ax-6%2F5%2B%283%2F5%29%2Ax=17 Reduce any fractions

2%2Ax%2B%283%2F5%29%2Ax=17%2B6%2F5Add 6%2F5 to both sides


2%2Ax%2B%283%2F5%29%2Ax=85%2F5%2B6%2F5 Make 17 into a fraction with a denominator of 5


2%2Ax%2B%283%2F5%29%2Ax=91%2F5 Combine the terms on the right side



%2810%2F5%29%2Ax%2B%283%2F5%29x=91%2F5 Make 2 into a fraction with a denominator of 5

%2813%2F5%29%2Ax=91%2F5 Now combine the terms on the left side.


cross%28%285%2F13%29%2813%2F5%29%29x=%2891%2F5%29%285%2F13%29 Multiply both sides by 5%2F13. This will cancel out 13%2F5 and isolate x

So when we multiply 91%2F5 and 5%2F13 (and simplify) we get



x=7 <---------------------------------One answer

Now that we know that x=7, lets substitute that in for x to solve for y

2%287%29%2B3%2Ay=17 Plug in x=7 into the 2nd equation

14%2B3%2Ay=17 Multiply

3%2Ay=17-14Subtract 14 from both sides

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

cross%28%281%2F3%29%283%29%29%2Ay=%283%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=7 and y=1

which can also look like

(7,1)

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

1%2Ax-5%2Ay=2
2%2Ax%2B3%2Ay=17

we get


graph of 1%2Ax-5%2Ay=2 (red) and 2%2Ax%2B3%2Ay=17 (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 (7,1). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (7,1) into the system of equations


Let x=7 and y=1. Now plug those values into the equation 1%2Ax-5%2Ay=2

1%2A%287%29-5%2A%281%29=2 Plug in x=7 and y=1


7-5=2 Multiply


2=2 Add


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


So the solution (7,1) satisfies 1%2Ax-5%2Ay=2



Let x=7 and y=1. Now plug those values into the equation 2%2Ax%2B3%2Ay=17

2%2A%287%29%2B3%2A%281%29=17 Plug in x=7 and y=1


14%2B3=17 Multiply


17=17 Add


17=17 Reduce. Since this equation is true the solution works.


So the solution (7,1) satisfies 2%2Ax%2B3%2Ay=17


Since the solution (7,1) satisfies the system of equations


1%2Ax-5%2Ay=2
2%2Ax%2B3%2Ay=17


this verifies our answer.