SOLUTION: solve by the substitution method x+3y=-8,4x-3y=23

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Question 96755: solve by the substitution method x+3y=-8,4x-3y=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

1%2Ax%2B3%2Ay=-8
4%2Ax-3%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

3%2Ay=-8-1%2AxSubtract 1%2Ax from both sides

y=%28-8-1%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=-8%2F3-%281%2F3%29%2Ax Now we've fully isolated y

Since y equals -8%2F3-%281%2F3%29%2Ax we can substitute the expression -8%2F3-%281%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B-3%2Ahighlight%28%28-8%2F3-%281%2F3%29%2Ax%29%29=23 Replace y with -8%2F3-%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax-3%2A%28-8%2F3%29-3%28-1%2F3%29x=23 Distribute -3 to -8%2F3-%281%2F3%29%2Ax

4%2Ax%2B24%2F3%2B%283%2F3%29%2Ax=23 Multiply



4%2Ax%2B8%2B1%2Ax=23 Reduce any fractions

4%2Ax%2B1%2Ax=23-8 Subtract 8 from both sides


4%2Ax%2B1%2Ax=15 Combine the terms on the right side



5%2Ax=15 Now combine the terms on the left side.


cross%28%281%2F5%29%285%2F1%29%29x=%2815%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5%2F1 and isolate x

So when we multiply 15%2F1 and 1%2F5 (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

4%283%29-3%2Ay=23 Plug in x=3 into the 2nd equation

12-3%2Ay=23 Multiply

-3%2Ay=23-12Subtract 12 from both sides

-3%2Ay=11 Combine the terms on the right side

cross%28%281%2F-3%29%28-3%29%29%2Ay=%2811%2F1%29%281%2F-3%29 Multiply both sides by 1%2F-3. This will cancel out -3 on the left side.

y=11%2F-3 Multiply the terms on the right side


y=-11%2F3 Reduce


So this is the other answer


y=-11%2F3<---------------------------------Other answer


So our solution is

x=3 and y=-11%2F3

which can also look like

(3,-11%2F3)

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

1%2Ax%2B3%2Ay=-8
4%2Ax-3%2Ay=23

we get


graph of 1%2Ax%2B3%2Ay=-8 (red) and 4%2Ax-3%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 (3,-11%2F3). This verifies our answer.


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Check:

Plug in (3,-11%2F3) into the system of equations


Let x=3 and y=-11%2F3. Now plug those values into the equation 1%2Ax%2B3%2Ay=-8

1%2A%283%29%2B3%2A%28-11%2F3%29=-8 Plug in x=3 and y=-11%2F3


3-33%2F3=-8 Multiply


-24%2F3=-8 Add


-8=-8 Reduce. Since this equation is true the solution works.


So the solution (3,-11%2F3) satisfies 1%2Ax%2B3%2Ay=-8



Let x=3 and y=-11%2F3. Now plug those values into the equation 4%2Ax-3%2Ay=23

4%2A%283%29-3%2A%28-11%2F3%29=23 Plug in x=3 and y=-11%2F3


12%2B33%2F3=23 Multiply


69%2F3=23 Add


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


So the solution (3,-11%2F3) satisfies 4%2Ax-3%2Ay=23


Since the solution (3,-11%2F3) satisfies the system of equations


1%2Ax%2B3%2Ay=-8
4%2Ax-3%2Ay=23


this verifies our answer.