SOLUTION: Solve each system of equations. Check by substitution 5x + 3t = 19 5x - 3t = 1

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Question 120285: Solve each system of equations. Check by substitution
5x + 3t = 19
5x - 3t = 1

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
5x+%2B+3t+=+19
5x+-+3t+=++1.............let t=y
then you have:
5x+%2B+3y+=+19
5x+-+3y+=++1
solution:

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


Lets start with the given system of linear equations

5%2Ax%2B3%2Ay=19
5%2Ax-3%2Ay=1

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

3%2Ay=19-5%2AxSubtract 5%2Ax from both sides

y=%2819-5%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=19%2F3-%285%2F3%29%2Ax Now we've fully isolated y

Since y equals 19%2F3-%285%2F3%29%2Ax we can substitute the expression 19%2F3-%285%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


5%2Ax%2B-3%2Ahighlight%28%2819%2F3-%285%2F3%29%2Ax%29%29=1 Replace y with 19%2F3-%285%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

5%2Ax-3%2A%2819%2F3%29-3%28-5%2F3%29x=1 Distribute -3 to 19%2F3-%285%2F3%29%2Ax

5%2Ax-57%2F3%2B%2815%2F3%29%2Ax=1 Multiply



5%2Ax-19%2B5%2Ax=1 Reduce any fractions

5%2Ax%2B5%2Ax=1%2B19Add 19 to both sides


5%2Ax%2B5%2Ax=20 Combine the terms on the right side



10%2Ax=20 Now combine the terms on the left side.


cross%28%281%2F10%29%2810%2F1%29%29x=%2820%2F1%29%281%2F10%29 Multiply both sides by 1%2F10. This will cancel out 10%2F1 and isolate x

So when we multiply 20%2F1 and 1%2F10 (and simplify) we get



x=2 <---------------------------------One answer

Now that we know that x=2, lets substitute that in for x to solve for y

5%282%29-3%2Ay=1 Plug in x=2 into the 2nd equation

10-3%2Ay=1 Multiply

-3%2Ay=1-10Subtract 10 from both sides

-3%2Ay=-9 Combine the terms on the right side

cross%28%281%2F-3%29%28-3%29%29%2Ay=%28-9%2F1%29%281%2F-3%29 Multiply both sides by 1%2F-3. This will cancel out -3 on the left side.

y=-9%2F-3 Multiply the terms on the right side


y=3 Reduce


So this is the other answer


y=3<---------------------------------Other answer


So our solution is

x=2 and y=3

which can also look like

(2,3)

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

5%2Ax%2B3%2Ay=19
5%2Ax-3%2Ay=1

we get


graph of 5%2Ax%2B3%2Ay=19 (red) and 5%2Ax-3%2Ay=1 (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 (2,3). This verifies our answer.


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Check:

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


Let x=2 and y=3. Now plug those values into the equation 5%2Ax%2B3%2Ay=19

5%2A%282%29%2B3%2A%283%29=19 Plug in x=2 and y=3


10%2B9=19 Multiply


19=19 Add


19=19 Reduce. Since this equation is true the solution works.


So the solution (2,3) satisfies 5%2Ax%2B3%2Ay=19



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

5%2A%282%29-3%2A%283%29=1 Plug in x=2 and y=3


10-9=1 Multiply


1=1 Add


1=1 Reduce. Since this equation is true the solution works.


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


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


5%2Ax%2B3%2Ay=19
5%2Ax-3%2Ay=1


this verifies our answer.