SOLUTION: Solve the system by substitution. x – 4y = –19 –4x – 5y = 13

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Question 82387: Solve the system by substitution.
x – 4y = –19
–4x – 5y = 13
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-4%2Ay=-19
-4%2Ax-5%2Ay=13

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

-4%2Ay=-19-1%2AxSubtract 1%2Ax from both sides

y=%28-19-1%2Ax%29%2F-4 Divide both sides by -4.


Which breaks down and reduces to



y=19%2F4%2B%281%2F4%29%2Ax Now we've fully isolated y

Since y equals 19%2F4%2B%281%2F4%29%2Ax we can substitute the expression 19%2F4%2B%281%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-4%2Ax%2B-5%2Ahighlight%28%2819%2F4%2B%281%2F4%29%2Ax%29%29=13 Replace y with 19%2F4%2B%281%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

-4%2Ax-5%2A%2819%2F4%29-5%281%2F4%29x=13 Distribute -5 to 19%2F4%2B%281%2F4%29%2Ax

-4%2Ax-95%2F4-%285%2F4%29%2Ax=13 Multiply



-4%2Ax-95%2F4-%285%2F4%29%2Ax=13 Reduce any fractions

-4%2Ax-%285%2F4%29%2Ax=13%2B95%2F4Add 95%2F4 to both sides


-4%2Ax-%285%2F4%29%2Ax=52%2F4%2B95%2F4 Make 13 into a fraction with a denominator of 4


-4%2Ax-%285%2F4%29%2Ax=147%2F4 Combine the terms on the right side



%28-16%2F4%29%2Ax-%285%2F4%29x=147%2F4 Make -4 into a fraction with a denominator of 4

%28-21%2F4%29%2Ax=147%2F4 Now combine the terms on the left side.


cross%28%284%2F-21%29%28-21%2F4%29%29x=%28147%2F4%29%284%2F-21%29 Multiply both sides by 4%2F-21. This will cancel out -21%2F4 and isolate x

So when we multiply 147%2F4 and 4%2F-21 (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

-4%28-7%29-5%2Ay=13 Plug in x=-7 into the 2nd equation

28-5%2Ay=13 Multiply

-5%2Ay=13-28Subtract 28 from both sides

-5%2Ay=-15 Combine the terms on the right side

cross%28%281%2F-5%29%28-5%29%29%2Ay=%28-15%2F1%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5 on the left side.

y=-15%2F-5 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=-7 and y=3

which can also look like

(-7,3)

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

1%2Ax-4%2Ay=-19
-4%2Ax-5%2Ay=13

we get


graph of 1%2Ax-4%2Ay=-19 (red) and -4%2Ax-5%2Ay=13 (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 (-7,3). This verifies our answer.


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Check:

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


Let x=-7 and y=3. Now plug those values into the equation 1%2Ax-4%2Ay=-19

1%2A%28-7%29-4%2A%283%29=-19 Plug in x=-7 and y=3


-7-12=-19 Multiply


-19=-19 Add


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


So the solution (-7,3) satisfies 1%2Ax-4%2Ay=-19



Let x=-7 and y=3. Now plug those values into the equation -4%2Ax-5%2Ay=13

-4%2A%28-7%29-5%2A%283%29=13 Plug in x=-7 and y=3


28-15=13 Multiply


13=13 Add


13=13 Reduce. Since this equation is true the solution works.


So the solution (-7,3) satisfies -4%2Ax-5%2Ay=13


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


1%2Ax-4%2Ay=-19
-4%2Ax-5%2Ay=13


this verifies our answer.