SOLUTION: If I have 5x+y=9 over 10x-7y=-18, what formula can be used?

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Question 699924: If I have 5x+y=9 over 10x-7y=-18, what formula can be used?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

5x%2By=9 .........1
+10x-7y=-18.....2
______________________....if you have this, than you have a system of linear equations and you can solve it by graphing, by substitution or by addition (elimination)
-by graphing

Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


5x%2By=9

10x-7y=-18





In order to graph these equations, we need to solve for y for each equation.




So let's solve for y on the first equation


5x%2By=9 Start with the given equation



1y=9-5x Subtract 5+x from both sides



1y=-5x%2B9 Rearrange the equation



y=%28-5x%2B9%29%2F%281%29 Divide both sides by 1



y=%28-5%2F1%29x%2B%289%29%2F%281%29 Break up the fraction



y=-5x%2B9 Reduce



Now lets graph y=-5x%2B9 (note: if you need help with graphing, check out this solver)



+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-5x%2B9%29+ Graph of y=-5x%2B9




So let's solve for y on the second equation


10x-7y=-18 Start with the given equation



-7y=-18-10x Subtract 10+x from both sides



-7y=-10x-18 Rearrange the equation



y=%28-10x-18%29%2F%28-7%29 Divide both sides by -7



y=%28-10%2F-7%29x%2B%28-18%29%2F%28-7%29 Break up the fraction



y=%2810%2F7%29x%2B18%2F7 Reduce





Now lets add the graph of y=%2810%2F7%29x%2B18%2F7 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-5x%2B9%2C%2810%2F7%29x%2B18%2F7%29+ Graph of y=-5x%2B9(red) and y=%2810%2F7%29x%2B18%2F7(green)


From the graph, we can see that the two lines intersect at the point (1,4) (note: you might have to adjust the window to see the intersection)



-by substitution

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


Lets start with the given system of linear equations

5%2Ax%2B1%2Ay=9
10%2Ax-7%2Ay=-18

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

1%2Ay=9-5%2AxSubtract 5%2Ax from both sides

y=%289-5%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=9-5%2Ax Now we've fully isolated y

Since y equals 9-5%2Ax we can substitute the expression 9-5%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


10%2Ax%2B-7%2Ahighlight%28%289-5%2Ax%29%29=-18 Replace y with 9-5%2Ax. Since this eliminates y, we can now solve for x.

10%2Ax-7%2A%289%29-7%28-5%29x=-18 Distribute -7 to 9-5%2Ax

10%2Ax-63%2B35%2Ax=-18 Multiply



10%2Ax-63%2B35%2Ax=-18 Reduce any fractions

10%2Ax%2B35%2Ax=-18%2B63Add 63 to both sides


10%2Ax%2B35%2Ax=45 Combine the terms on the right side



45%2Ax=45 Now combine the terms on the left side.


cross%28%281%2F45%29%2845%2F1%29%29x=%2845%2F1%29%281%2F45%29 Multiply both sides by 1%2F45. This will cancel out 45%2F1 and isolate x

So when we multiply 45%2F1 and 1%2F45 (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

10%281%29-7%2Ay=-18 Plug in x=1 into the 2nd equation

10-7%2Ay=-18 Multiply

-7%2Ay=-18-10Subtract 10 from both sides

-7%2Ay=-28 Combine the terms on the right side

cross%28%281%2F-7%29%28-7%29%29%2Ay=%28-28%2F1%29%281%2F-7%29 Multiply both sides by 1%2F-7. This will cancel out -7 on the left side.

y=-28%2F-7 Multiply the terms on the right side


y=4 Reduce


So this is the other answer


y=4<---------------------------------Other answer


So our solution is

x=1 and y=4

which can also look like

(1,4)

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

5%2Ax%2B1%2Ay=9
10%2Ax-7%2Ay=-18

we get


graph of 5%2Ax%2B1%2Ay=9 (red) and 10%2Ax-7%2Ay=-18 (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 (1,4). This verifies our answer.


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Check:

Plug in (1,4) into the system of equations


Let x=1 and y=4. Now plug those values into the equation 5%2Ax%2B1%2Ay=9

5%2A%281%29%2B1%2A%284%29=9 Plug in x=1 and y=4


5%2B4=9 Multiply


9=9 Add


9=9 Reduce. Since this equation is true the solution works.


So the solution (1,4) satisfies 5%2Ax%2B1%2Ay=9



Let x=1 and y=4. Now plug those values into the equation 10%2Ax-7%2Ay=-18

10%2A%281%29-7%2A%284%29=-18 Plug in x=1 and y=4


10-28=-18 Multiply


-18=-18 Add


-18=-18 Reduce. Since this equation is true the solution works.


So the solution (1,4) satisfies 10%2Ax-7%2Ay=-18


Since the solution (1,4) satisfies the system of equations


5%2Ax%2B1%2Ay=9
10%2Ax-7%2Ay=-18


this verifies our answer.





-by addition (elimination)

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


Lets start with the given system of linear equations

5%2Ax%2B1%2Ay=9
10%2Ax-7%2Ay=-18

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 5 and 10 to some equal number, we could try to get them to the LCM.

Since the LCM of 5 and 10 is 10, 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%285%2Ax%2B1%2Ay%29=%289%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%2810%2Ax-7%2Ay%29=%28-18%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
10%2Ax%2B2%2Ay=18
-10%2Ax%2B7%2Ay=18

Notice how 10 and -10 add to zero (ie 10%2B-10=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2810%2Ax-10%2Ax%29%2B%282%2Ay%2B7%2Ay%29=18%2B18

%2810-10%29%2Ax%2B%282%2B7%29y=18%2B18

cross%2810%2B-10%29%2Ax%2B%282%2B7%29%2Ay=18%2B18 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:

9%2Ay=36

y=36%2F9 Divide both sides by 9 to solve for y



y=4 Reduce


Now plug this answer into the top equation 5%2Ax%2B1%2Ay=9 to solve for x

5%2Ax%2B1%284%29=9 Plug in y=4


5%2Ax%2B4=9 Multiply



5%2Ax=9-4 Subtract 4 from both sides

5%2Ax=5 Combine the terms on the right side

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


x=1 Multiply the terms on the right side


So our answer is

x=1, y=4

which also looks like

(1, 4)

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

5%2Ax%2B1%2Ay=9
10%2Ax-7%2Ay=-18

we get



graph of 5%2Ax%2B1%2Ay=9 (red) 10%2Ax-7%2Ay=-18 (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 (1,4). This verifies our answer.