SOLUTION: Solve each system: 8x-3y=3 3x-2y+5=0

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Question 115035This question is from textbook
: Solve each system:
8x-3y=3
3x-2y+5=0 This question is from textbook

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

8%2Ax-3%2Ay=3
3%2Ax-2%2Ay=-5

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=3-8%2AxSubtract 8%2Ax from both sides

y=%283-8%2Ax%29%2F-3 Divide both sides by -3.


Which breaks down and reduces to



y=-1%2B%288%2F3%29%2Ax Now we've fully isolated y

Since y equals -1%2B%288%2F3%29%2Ax we can substitute the expression -1%2B%288%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-2%2Ahighlight%28%28-1%2B%288%2F3%29%2Ax%29%29=-5 Replace y with -1%2B%288%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-2%2A%28-1%29-2%288%2F3%29x=-5 Distribute -2 to -1%2B%288%2F3%29%2Ax

3%2Ax%2B2-%2816%2F3%29%2Ax=-5 Multiply



3%2Ax%2B2-%2816%2F3%29%2Ax=-5 Reduce any fractions

3%2Ax-%2816%2F3%29%2Ax=-5-2 Subtract 2 from both sides


3%2Ax-%2816%2F3%29%2Ax=-7 Combine the terms on the right side



%289%2F3%29%2Ax-%2816%2F3%29x=-7 Make 3 into a fraction with a denominator of 3

%28-7%2F3%29%2Ax=-7 Now combine the terms on the left side.


cross%28%283%2F-7%29%28-7%2F3%29%29x=%28-7%2F1%29%283%2F-7%29 Multiply both sides by 3%2F-7. This will cancel out -7%2F3 and isolate x

So when we multiply -7%2F1 and 3%2F-7 (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

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

9-2%2Ay=-5 Multiply

-2%2Ay=-5-9Subtract 9 from both sides

-2%2Ay=-14 Combine the terms on the right side

cross%28%281%2F-2%29%28-2%29%29%2Ay=%28-14%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=-14%2F-2 Multiply the terms on the right side


y=7 Reduce


So this is the other answer


y=7<---------------------------------Other answer


So our solution is

x=3 and y=7

which can also look like

(3,7)

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

8%2Ax-3%2Ay=3
3%2Ax-2%2Ay=-5

we get


graph of 8%2Ax-3%2Ay=3 (red) and 3%2Ax-2%2Ay=-5 (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 (3,7). This verifies our answer.


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Check:

Plug in (3,7) into the system of equations


Let x=3 and y=7. Now plug those values into the equation 8%2Ax-3%2Ay=3

8%2A%283%29-3%2A%287%29=3 Plug in x=3 and y=7


24-21=3 Multiply


3=3 Add


3=3 Reduce. Since this equation is true the solution works.


So the solution (3,7) satisfies 8%2Ax-3%2Ay=3



Let x=3 and y=7. Now plug those values into the equation 3%2Ax-2%2Ay=-5

3%2A%283%29-2%2A%287%29=-5 Plug in x=3 and y=7


9-14=-5 Multiply


-5=-5 Add


-5=-5 Reduce. Since this equation is true the solution works.


So the solution (3,7) satisfies 3%2Ax-2%2Ay=-5


Since the solution (3,7) satisfies the system of equations


8%2Ax-3%2Ay=3
3%2Ax-2%2Ay=-5


this verifies our answer.