SOLUTION: Which is the best method for solving the system? graphing table elimination or substitution 9x+8y=7 18x-15y=14

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Question 950805: Which is the best method for solving the system?
graphing table elimination or substitution
9x+8y=7
18x-15y=14
Found 2 solutions by Alan3354, MathLover1:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Which is the best method for solving the system?
graphing table elimination or substitution
9x+8y=7
18x-15y=14
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"Best" is not a clearly defined math term.
I would multiply the 1st eqn by 2, then subtract (elimination).
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If it's more than 2 eqns and variables I would use determinants.

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
9x%2B8y=7
18x-15y=14
1.
to solve by graphing, make table for each function, choose two values for x, calculate y, then plot points and draw lines; the point where these lines intersect will be solution to this system
9x%2B8y=7....first solve for y
y=-%289%2F8%29x%2B7%2F8
x|y
0|7%2F8
7%2F9|0

18x-15y=14
18x-14=15y
%2818%2F15%29x-14%2F15=y
%286%2F5%29x-14%2F15=y
x|y
0|-14%2F15
7%2F9|0


the two lines intersect at the point (7%2F9,0)

2.
elimination
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

9%2Ax%2B8%2Ay=7
18%2Ax-15%2Ay=14

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 9 and 18 to some equal number, we could try to get them to the LCM.

Since the LCM of 9 and 18 is 18, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:

2%2A%289%2Ax%2B8%2Ay%29=%287%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%2818%2Ax-15%2Ay%29=%2814%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
18%2Ax%2B16%2Ay=14
-18%2Ax%2B15%2Ay=-14

Notice how 18 and -18 add to zero (ie 18%2B-18=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2818%2Ax-18%2Ax%29%2B%2816%2Ay%2B15%2Ay%29=14-14

%2818-18%29%2Ax%2B%2816%2B15%29y=14-14

cross%2818%2B-18%29%2Ax%2B%2816%2B15%29%2Ay=14-14 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:

31%2Ay=0

y=0%2F31 Divide both sides by 31 to solve for y



y=0 Reduce


Now plug this answer into the top equation 9%2Ax%2B8%2Ay=7 to solve for x

9%2Ax%2B8%280%29=7 Plug in y=0


9%2Ax%2B0=7 Multiply



9%2Ax=7-0 Subtract 0 from both sides

9%2Ax=7 Combine the terms on the right side

cross%28%281%2F9%29%289%29%29%2Ax=%287%29%281%2F9%29 Multiply both sides by 1%2F9. This will cancel out 9 on the left side.


x=7%2F9 Multiply the terms on the right side


So our answer is

x=7%2F9, y=0

which also looks like

(7%2F9, 0)

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

9%2Ax%2B8%2Ay=7
18%2Ax-15%2Ay=14

we get



graph of 9%2Ax%2B8%2Ay=7 (red) 18%2Ax-15%2Ay=14 (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 (7%2F9,0). This verifies our answer.


3.
substitution
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

9%2Ax%2B8%2Ay=7
18%2Ax-15%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

8%2Ay=7-9%2AxSubtract 9%2Ax from both sides

y=%287-9%2Ax%29%2F8 Divide both sides by 8.


Which breaks down and reduces to



y=7%2F8-%289%2F8%29%2Ax Now we've fully isolated y

Since y equals 7%2F8-%289%2F8%29%2Ax we can substitute the expression 7%2F8-%289%2F8%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


18%2Ax%2B-15%2Ahighlight%28%287%2F8-%289%2F8%29%2Ax%29%29=14 Replace y with 7%2F8-%289%2F8%29%2Ax. Since this eliminates y, we can now solve for x.

18%2Ax-15%2A%287%2F8%29-15%28-9%2F8%29x=14 Distribute -15 to 7%2F8-%289%2F8%29%2Ax

18%2Ax-105%2F8%2B%28135%2F8%29%2Ax=14 Multiply



18%2Ax-105%2F8%2B%28135%2F8%29%2Ax=14 Reduce any fractions

18%2Ax%2B%28135%2F8%29%2Ax=14%2B105%2F8Add 105%2F8 to both sides


18%2Ax%2B%28135%2F8%29%2Ax=112%2F8%2B105%2F8 Make 14 into a fraction with a denominator of 8


18%2Ax%2B%28135%2F8%29%2Ax=217%2F8 Combine the terms on the right side



%28144%2F8%29%2Ax%2B%28135%2F8%29x=217%2F8 Make 18 into a fraction with a denominator of 8

%28279%2F8%29%2Ax=217%2F8 Now combine the terms on the left side.


cross%28%288%2F279%29%28279%2F8%29%29x=%28217%2F8%29%288%2F279%29 Multiply both sides by 8%2F279. This will cancel out 279%2F8 and isolate x

So when we multiply 217%2F8 and 8%2F279 (and simplify) we get



x=7%2F9 <---------------------------------One answer

Now that we know that x=7%2F9, lets substitute that in for x to solve for y

18%287%2F9%29-15%2Ay=14 Plug in x=7%2F9 into the 2nd equation

14-15%2Ay=14 Multiply

-15%2Ay=14-14Subtract 14 from both sides

-15%2Ay=0 Combine the terms on the right side

cross%28%281%2F-15%29%28-15%29%29%2Ay=%280%2F1%29%281%2F-15%29 Multiply both sides by 1%2F-15. This will cancel out -15 on the left side.

y=0%2F-15 Multiply the terms on the right side


y=0 Reduce


So this is the other answer


y=0<---------------------------------Other answer


So our solution is

x=7%2F9 and y=0

which can also look like

(7%2F9,0)

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

9%2Ax%2B8%2Ay=7
18%2Ax-15%2Ay=14

we get


graph of 9%2Ax%2B8%2Ay=7 (red) and 18%2Ax-15%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%2F9,0). This verifies our answer.


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Check:

Plug in (7%2F9,0) into the system of equations


Let x=7%2F9 and y=0. Now plug those values into the equation 9%2Ax%2B8%2Ay=7

9%2A%287%2F9%29%2B8%2A%280%29=7 Plug in x=7%2F9 and y=0


63%2F9%2B0=7 Multiply


63%2F9=7 Add


7=7 Reduce. Since this equation is true the solution works.


So the solution (7%2F9,0) satisfies 9%2Ax%2B8%2Ay=7



Let x=7%2F9 and y=0. Now plug those values into the equation 18%2Ax-15%2Ay=14

18%2A%287%2F9%29-15%2A%280%29=14 Plug in x=7%2F9 and y=0


126%2F9%2B0=14 Multiply


126%2F9=14 Add


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


So the solution (7%2F9,0) satisfies 18%2Ax-15%2Ay=14


Since the solution (7%2F9,0) satisfies the system of equations


9%2Ax%2B8%2Ay=7
18%2Ax-15%2Ay=14


this verifies our answer.





Which is the best method for solving the system?
Depends on given equations. There are pros and cons for each method like:

Graphing:
Pro: gives you a visual picture of what's going on and is good for really big numbers
Con: if your graphing skills are not accurate or the solution is not at the intersection of a grid line, you have to estimate your answer.
Substitution:
Pro: Gives you an exact answer
Con: Sloppy calculations can cause incorrect answers and if one equation is not already solved for a variable (y = or x = ), you have to do extra work
Elimination
Pro: Works for any system of linear equations. After one or two steps you have a one-step equation to solve and you have half of your solution.
Con: Sloppy calculations can lead to incorrect solutions
If I need a rough idea of the solution, I graph.
If one of the equations is already solved for y or for+x, I use substitution.
If neither of the above is true, I use elimination.
In this case, I prefer graphing.