SOLUTION: solve each systems of equations algebraically 28. {2x+4y=10} {3x-4y=-15}

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Question 120659: solve each systems of equations algebraically
28. {2x+4y=10}
{3x-4y=-15}
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

2%2Ax%2B4%2Ay=10
3%2Ax-4%2Ay=-15

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=10-2%2AxSubtract 2%2Ax from both sides

y=%2810-2%2Ax%29%2F4 Divide both sides by 4.


Which breaks down and reduces to



y=5%2F2-%281%2F2%29%2Ax Now we've fully isolated y

Since y equals 5%2F2-%281%2F2%29%2Ax we can substitute the expression 5%2F2-%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-4%2Ahighlight%28%285%2F2-%281%2F2%29%2Ax%29%29=-15 Replace y with 5%2F2-%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-4%2A%285%2F2%29-4%28-1%2F2%29x=-15 Distribute -4 to 5%2F2-%281%2F2%29%2Ax

3%2Ax-20%2F2%2B%284%2F2%29%2Ax=-15 Multiply



3%2Ax-10%2B2%2Ax=-15 Reduce any fractions

3%2Ax%2B2%2Ax=-15%2B10Add 10 to both sides


3%2Ax%2B2%2Ax=-5 Combine the terms on the right side



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


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

So when we multiply -5%2F1 and 1%2F5 (and simplify) we get



x=-1 <---------------------------------One answer

Now that we know that x=-1, lets substitute that in for x to solve for y

3%28-1%29-4%2Ay=-15 Plug in x=-1 into the 2nd equation

-3-4%2Ay=-15 Multiply

-4%2Ay=-15%2B3Add 3 to both sides

-4%2Ay=-12 Combine the terms on the right side

cross%28%281%2F-4%29%28-4%29%29%2Ay=%28-12%2F1%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 on the left side.

y=-12%2F-4 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=-1 and y=3

which can also look like

(-1,3)

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

2%2Ax%2B4%2Ay=10
3%2Ax-4%2Ay=-15

we get


graph of 2%2Ax%2B4%2Ay=10 (red) and 3%2Ax-4%2Ay=-15 (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 (-1,3). This verifies our answer.


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Check:

Plug in (-1,3) into the system of equations


Let x=-1 and y=3. Now plug those values into the equation 2%2Ax%2B4%2Ay=10

2%2A%28-1%29%2B4%2A%283%29=10 Plug in x=-1 and y=3


-2%2B12=10 Multiply


10=10 Add


10=10 Reduce. Since this equation is true the solution works.


So the solution (-1,3) satisfies 2%2Ax%2B4%2Ay=10



Let x=-1 and y=3. Now plug those values into the equation 3%2Ax-4%2Ay=-15

3%2A%28-1%29-4%2A%283%29=-15 Plug in x=-1 and y=3


-3-12=-15 Multiply


-15=-15 Add


-15=-15 Reduce. Since this equation is true the solution works.


So the solution (-1,3) satisfies 3%2Ax-4%2Ay=-15


Since the solution (-1,3) satisfies the system of equations


2%2Ax%2B4%2Ay=10
3%2Ax-4%2Ay=-15


this verifies our answer.