SOLUTION: how do you solve using elimination and substitution..i have 29 problems to answer so this helps me umm here is the problem 2x-7y=8 and 3x-4y=-1

Algebra ->  Expressions-with-variables -> SOLUTION: how do you solve using elimination and substitution..i have 29 problems to answer so this helps me umm here is the problem 2x-7y=8 and 3x-4y=-1      Log On


   

Question 120453: how do you solve using elimination and substitution..i have 29 problems to answer so this helps me umm here is the problem 2x-7y=8 and 3x-4y=-1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Substitution:

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax-7%2Ay=8
3%2Ax-4%2Ay=-1

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

-7%2Ay=8-2%2AxSubtract 2%2Ax from both sides

y=%288-2%2Ax%29%2F-7 Divide both sides by -7.


Which breaks down and reduces to



y=-8%2F7%2B%282%2F7%29%2Ax Now we've fully isolated y

Since y equals -8%2F7%2B%282%2F7%29%2Ax we can substitute the expression -8%2F7%2B%282%2F7%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-4%2Ahighlight%28%28-8%2F7%2B%282%2F7%29%2Ax%29%29=-1 Replace y with -8%2F7%2B%282%2F7%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-4%2A%28-8%2F7%29-4%282%2F7%29x=-1 Distribute -4 to -8%2F7%2B%282%2F7%29%2Ax

3%2Ax%2B32%2F7-%288%2F7%29%2Ax=-1 Multiply



3%2Ax%2B32%2F7-%288%2F7%29%2Ax=-1 Reduce any fractions

3%2Ax-%288%2F7%29%2Ax=-1-32%2F7 Subtract 32%2F7 from both sides


3%2Ax-%288%2F7%29%2Ax=-7%2F7-32%2F7 Make -1 into a fraction with a denominator of 7


3%2Ax-%288%2F7%29%2Ax=-39%2F7 Combine the terms on the right side



%2821%2F7%29%2Ax-%288%2F7%29x=-39%2F7 Make 3 into a fraction with a denominator of 7

%2813%2F7%29%2Ax=-39%2F7 Now combine the terms on the left side.


cross%28%287%2F13%29%2813%2F7%29%29x=%28-39%2F7%29%287%2F13%29 Multiply both sides by 7%2F13. This will cancel out 13%2F7 and isolate x

So when we multiply -39%2F7 and 7%2F13 (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%28-3%29-4%2Ay=-1 Plug in x=-3 into the 2nd equation

-9-4%2Ay=-1 Multiply

-4%2Ay=-1%2B9Add 9 to both sides

-4%2Ay=8 Combine the terms on the right side

cross%28%281%2F-4%29%28-4%29%29%2Ay=%288%2F1%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 on the left side.

y=8%2F-4 Multiply the terms on the right side


y=-2 Reduce


So this is the other answer


y=-2<---------------------------------Other answer


So our solution is

x=-3 and y=-2

which can also look like

(-3,-2)

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

2%2Ax-7%2Ay=8
3%2Ax-4%2Ay=-1

we get


graph of 2%2Ax-7%2Ay=8 (red) and 3%2Ax-4%2Ay=-1 (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,-2). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

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


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

2%2A%28-3%29-7%2A%28-2%29=8 Plug in x=-3 and y=-2


-6%2B14=8 Multiply


8=8 Add


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


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



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

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


-9%2B8=-1 Multiply


-1=-1 Add


-1=-1 Reduce. Since this equation is true the solution works.


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


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


2%2Ax-7%2Ay=8
3%2Ax-4%2Ay=-1


this verifies our answer.








Elimination:

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

2%2Ax-7%2Ay=8
3%2Ax-4%2Ay=-1

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 2 and 3 to some equal number, we could try to get them to the LCM.

Since the LCM of 2 and 3 is 6, we need to multiply both sides of the top equation by 3 and multiply both sides of the bottom equation by -2 like this:

3%2A%282%2Ax-7%2Ay%29=%288%29%2A3 Multiply the top equation (both sides) by 3
-2%2A%283%2Ax-4%2Ay%29=%28-1%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
6%2Ax-21%2Ay=24
-6%2Ax%2B8%2Ay=2

Notice how 6 and -6 add to zero (ie 6%2B-6=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%286%2Ax-6%2Ax%29-21%2Ay%2B8%2Ay%29=24%2B2

%286-6%29%2Ax-21%2B8%29y=24%2B2

cross%286%2B-6%29%2Ax%2B%28-21%2B8%29%2Ay=24%2B2 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-13%2Ay=26

y=26%2F-13 Divide both sides by -13 to solve for y



y=-2 Reduce


Now plug this answer into the top equation 2%2Ax-7%2Ay=8 to solve for x

2%2Ax-7%28-2%29=8 Plug in y=-2


2%2Ax%2B14=8 Multiply



2%2Ax=8-14 Subtract 14 from both sides

2%2Ax=-6 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ax=%28-6%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=-3 Multiply the terms on the right side


So our answer is

x=-3, y=-2

which also looks like

(-3, -2)

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

2%2Ax-7%2Ay=8
3%2Ax-4%2Ay=-1

we get



graph of 2%2Ax-7%2Ay=8 (red) 3%2Ax-4%2Ay=-1 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (-3,-2). This verifies our answer.