SOLUTION: I need help on understanding how to solve this problem using substitution and elimination. Please show me how: 4x+5y=13 4x+3y=9

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Question 655283: I need help on understanding how to solve this problem using substitution and elimination. Please show me how:
4x+5y=13
4x+3y=9
Found 2 solutions by MathLover1, ewatrrr:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

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


Lets start with the given system of linear equations

4%2Ax%2B5%2Ay=13
4%2Ax%2B3%2Ay=9

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

5%2Ay=13-4%2AxSubtract 4%2Ax from both sides

y=%2813-4%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=13%2F5-%284%2F5%29%2Ax Now we've fully isolated y

Since y equals 13%2F5-%284%2F5%29%2Ax we can substitute the expression 13%2F5-%284%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B3%2Ahighlight%28%2813%2F5-%284%2F5%29%2Ax%29%29=9 Replace y with 13%2F5-%284%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax%2B3%2A%2813%2F5%29%2B3%28-4%2F5%29x=9 Distribute 3 to 13%2F5-%284%2F5%29%2Ax

4%2Ax%2B39%2F5-%2812%2F5%29%2Ax=9 Multiply



4%2Ax%2B39%2F5-%2812%2F5%29%2Ax=9 Reduce any fractions

4%2Ax-%2812%2F5%29%2Ax=9-39%2F5 Subtract 39%2F5 from both sides


4%2Ax-%2812%2F5%29%2Ax=45%2F5-39%2F5 Make 9 into a fraction with a denominator of 5


4%2Ax-%2812%2F5%29%2Ax=6%2F5 Combine the terms on the right side



%2820%2F5%29%2Ax-%2812%2F5%29x=6%2F5 Make 4 into a fraction with a denominator of 5

%288%2F5%29%2Ax=6%2F5 Now combine the terms on the left side.


cross%28%285%2F8%29%288%2F5%29%29x=%286%2F5%29%285%2F8%29 Multiply both sides by 5%2F8. This will cancel out 8%2F5 and isolate x

So when we multiply 6%2F5 and 5%2F8 (and simplify) we get



x=3%2F4 <---------------------------------One answer

Now that we know that x=3%2F4, lets substitute that in for x to solve for y

4%283%2F4%29%2B3%2Ay=9 Plug in x=3%2F4 into the 2nd equation

3%2B3%2Ay=9 Multiply

3%2Ay=9-3Subtract 3 from both sides

3%2Ay=6 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ay=%286%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.

y=6%2F3 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%2F4 and y=2

which can also look like

(3%2F4,2)

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

4%2Ax%2B5%2Ay=13
4%2Ax%2B3%2Ay=9

we get


graph of 4%2Ax%2B5%2Ay=13 (red) and 4%2Ax%2B3%2Ay=9 (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%2F4,2). This verifies our answer.


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

Plug in (3%2F4,2) into the system of equations


Let x=3%2F4 and y=2. Now plug those values into the equation 4%2Ax%2B5%2Ay=13

4%2A%283%2F4%29%2B5%2A%282%29=13 Plug in x=3%2F4 and y=2


12%2F4%2B10=13 Multiply


52%2F4=13 Add


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


So the solution (3%2F4,2) satisfies 4%2Ax%2B5%2Ay=13



Let x=3%2F4 and y=2. Now plug those values into the equation 4%2Ax%2B3%2Ay=9

4%2A%283%2F4%29%2B3%2A%282%29=9 Plug in x=3%2F4 and y=2


12%2F4%2B6=9 Multiply


36%2F4=9 Add


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


So the solution (3%2F4,2) satisfies 4%2Ax%2B3%2Ay=9


Since the solution (3%2F4,2) satisfies the system of equations


4%2Ax%2B5%2Ay=13
4%2Ax%2B3%2Ay=9


this verifies our answer.






solving linear system by elimination:

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


Lets start with the given system of linear equations

4%2Ax%2B5%2Ay=13
4%2Ax%2B3%2Ay=9

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 4 and 4 to some equal number, we could try to get them to the LCM.

Since the LCM of 4 and 4 is 4, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -1 like this:

1%2A%284%2Ax%2B5%2Ay%29=%2813%29%2A1 Multiply the top equation (both sides) by 1
-1%2A%284%2Ax%2B3%2Ay%29=%289%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
4%2Ax%2B5%2Ay=13
-4%2Ax-3%2Ay=-9

Notice how 4 and -4 add to zero (ie 4%2B-4=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%284%2Ax-4%2Ax%29%2B%285%2Ay-3%2Ay%29=13-9

%284-4%29%2Ax%2B%285-3%29y=13-9

cross%284%2B-4%29%2Ax%2B%285-3%29%2Ay=13-9 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:

2%2Ay=4

y=4%2F2 Divide both sides by 2 to solve for y



y=2 Reduce


Now plug this answer into the top equation 4%2Ax%2B5%2Ay=13 to solve for x

4%2Ax%2B5%282%29=13 Plug in y=2


4%2Ax%2B10=13 Multiply



4%2Ax=13-10 Subtract 10 from both sides

4%2Ax=3 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ax=%283%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.


x=3%2F4 Multiply the terms on the right side


So our answer is

x=3%2F4, y=2

which also looks like

(3%2F4, 2)

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

4%2Ax%2B5%2Ay=13
4%2Ax%2B3%2Ay=9

we get



graph of 4%2Ax%2B5%2Ay=13 (red) 4%2Ax%2B3%2Ay=9 (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%2F4,2). This verifies our answer.

Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi,

Elimination:

4x+5y=13

4x+3y=9   | Subtracting 2nd from the 1st EQ to eliminate x

   2y = 4

    y = 2  and x = 3/4   4x+=+-5%2A2+%2B+13



substitution

 4x+5y=13   0r  4x = -5y +13

 4x+3y=9    0r  4x = -3y + 9

 -3y +9 = -5y + 13

  2y = 4

   y = 2   and x = 3/4