SOLUTION: Solve the system by substitution. x – 3y = 14 –3x + 5y = –14

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Question 111439: Solve the system by substitution.
x – 3y = 14
–3x + 5y = –14
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-3%2Ay=14
-3%2Ax%2B5%2Ay=-14

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=14-1%2AxSubtract 1%2Ax from both sides

y=%2814-1%2Ax%29%2F-3 Divide both sides by -3.


Which breaks down and reduces to



y=-14%2F3%2B%281%2F3%29%2Ax Now we've fully isolated y

Since y equals -14%2F3%2B%281%2F3%29%2Ax we can substitute the expression -14%2F3%2B%281%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-3%2Ax%2B5%2Ahighlight%28%28-14%2F3%2B%281%2F3%29%2Ax%29%29=-14 Replace y with -14%2F3%2B%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

-3%2Ax%2B5%2A%28-14%2F3%29%2B5%281%2F3%29x=-14 Distribute 5 to -14%2F3%2B%281%2F3%29%2Ax

-3%2Ax-70%2F3%2B%285%2F3%29%2Ax=-14 Multiply



-3%2Ax-70%2F3%2B%285%2F3%29%2Ax=-14 Reduce any fractions

-3%2Ax%2B%285%2F3%29%2Ax=-14%2B70%2F3Add 70%2F3 to both sides


-3%2Ax%2B%285%2F3%29%2Ax=-42%2F3%2B70%2F3 Make -14 into a fraction with a denominator of 3


-3%2Ax%2B%285%2F3%29%2Ax=28%2F3 Combine the terms on the right side



%28-9%2F3%29%2Ax%2B%285%2F3%29x=28%2F3 Make -3 into a fraction with a denominator of 3

%28-4%2F3%29%2Ax=28%2F3 Now combine the terms on the left side.


cross%28%283%2F-4%29%28-4%2F3%29%29x=%2828%2F3%29%283%2F-4%29 Multiply both sides by 3%2F-4. This will cancel out -4%2F3 and isolate x

So when we multiply 28%2F3 and 3%2F-4 (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

-3%28-7%29%2B5%2Ay=-14 Plug in x=-7 into the 2nd equation

21%2B5%2Ay=-14 Multiply

5%2Ay=-14-21Subtract 21 from both sides

5%2Ay=-35 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ay=%28-35%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.

y=-35%2F5 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=-7 and y=-7

which can also look like

(-7,-7)

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

1%2Ax-3%2Ay=14
-3%2Ax%2B5%2Ay=-14

we get


graph of 1%2Ax-3%2Ay=14 (red) and -3%2Ax%2B5%2Ay=-14 (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,-7). This verifies our answer.


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Check:

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


Let x=-7 and y=-7. Now plug those values into the equation 1%2Ax-3%2Ay=14

1%2A%28-7%29-3%2A%28-7%29=14 Plug in x=-7 and y=-7


-7%2B21=14 Multiply


14=14 Add


14=14 Reduce. Since this equation is true the solution works.


So the solution (-7,-7) satisfies 1%2Ax-3%2Ay=14



Let x=-7 and y=-7. Now plug those values into the equation -3%2Ax%2B5%2Ay=-14

-3%2A%28-7%29%2B5%2A%28-7%29=-14 Plug in x=-7 and y=-7


21-35=-14 Multiply


-14=-14 Add


-14=-14 Reduce. Since this equation is true the solution works.


So the solution (-7,-7) satisfies -3%2Ax%2B5%2Ay=-14


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


1%2Ax-3%2Ay=14
-3%2Ax%2B5%2Ay=-14


this verifies our answer.