SOLUTION: 6x-2y=11 x+3y=4 solve using substitution

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Question 120939:
6x-2y=11 x+3y=4
solve using substitution
Found 2 solutions by jim_thompson5910, checkley71:
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

6%2Ax-2%2Ay=11
1%2Ax%2B3%2Ay=4

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

-2%2Ay=11-6%2AxSubtract 6%2Ax from both sides

y=%2811-6%2Ax%29%2F-2 Divide both sides by -2.


Which breaks down and reduces to



y=-11%2F2%2B3%2Ax Now we've fully isolated y

Since y equals -11%2F2%2B3%2Ax we can substitute the expression -11%2F2%2B3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B3%2Ahighlight%28%28-11%2F2%2B3%2Ax%29%29=4 Replace y with -11%2F2%2B3%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B3%2A%28-11%2F2%29%2B3%283%29x=4 Distribute 3 to -11%2F2%2B3%2Ax

1%2Ax-33%2F2%2B9%2Ax=4 Multiply



1%2Ax-33%2F2%2B9%2Ax=4 Reduce any fractions

1%2Ax%2B9%2Ax=4%2B33%2F2Add 33%2F2 to both sides


1%2Ax%2B9%2Ax=8%2F2%2B33%2F2 Make 4 into a fraction with a denominator of 2


1%2Ax%2B9%2Ax=41%2F2 Combine the terms on the right side



10%2Ax=41%2F2 Now combine the terms on the left side.


cross%28%281%2F10%29%2810%2F1%29%29x=%2841%2F2%29%281%2F10%29 Multiply both sides by 1%2F10. This will cancel out 10%2F1 and isolate x

So when we multiply 41%2F2 and 1%2F10 (and simplify) we get



x=41%2F20 <---------------------------------One answer

Now that we know that x=41%2F20, lets substitute that in for x to solve for y

1%2841%2F20%29%2B3%2Ay=4 Plug in x=41%2F20 into the 2nd equation

41%2F20%2B3%2Ay=4 Multiply

3%2Ay=4-41%2F20Subtract 41%2F20 from both sides

3%2Ay=80%2F20-41%2F20 Make 4 into a fraction with a denominator of 20



3%2Ay=39%2F20 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ay=%2839%2F20%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.

y=39%2F60 Multiply the terms on the right side


y=13%2F20 Reduce


So this is the other answer


y=13%2F20<---------------------------------Other answer


So our solution is

x=41%2F20 and y=13%2F20

which can also look like

(41%2F20,13%2F20)

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

6%2Ax-2%2Ay=11
1%2Ax%2B3%2Ay=4

we get


graph of 6%2Ax-2%2Ay=11 (red) and 1%2Ax%2B3%2Ay=4 (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 (41%2F20,13%2F20). This verifies our answer.


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Check:

Plug in (41%2F20,13%2F20) into the system of equations


Let x=41%2F20 and y=13%2F20. Now plug those values into the equation 6%2Ax-2%2Ay=11

6%2A%2841%2F20%29-2%2A%2813%2F20%29=11 Plug in x=41%2F20 and y=13%2F20


246%2F20-26%2F20=11 Multiply


220%2F20=11 Add


11=11 Reduce. Since this equation is true the solution works.


So the solution (41%2F20,13%2F20) satisfies 6%2Ax-2%2Ay=11



Let x=41%2F20 and y=13%2F20. Now plug those values into the equation 1%2Ax%2B3%2Ay=4

1%2A%2841%2F20%29%2B3%2A%2813%2F20%29=4 Plug in x=41%2F20 and y=13%2F20


41%2F20%2B39%2F20=4 Multiply


80%2F20=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (41%2F20,13%2F20) satisfies 1%2Ax%2B3%2Ay=4


Since the solution (41%2F20,13%2F20) satisfies the system of equations


6%2Ax-2%2Ay=11
1%2Ax%2B3%2Ay=4


this verifies our answer.

Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
X+3Y=4 OR X=-3Y+4 NOW REPLACE X WITH (-3Y+4) IN THE OTHER EQUATION.
6(-3Y+4)-2Y=11
-18Y+24-2Y=11
-20Y=11-24
-20Y=-13
Y=-13/-20
Y=13/20 ANSWER.
X+3*13/20=4
X+39/20=4
X=4-39/20
X=(80-39)/20
X=41/20 ANSWER.
PROOF:
6*41/20-2*13/20=11
246/20-26/20=11
220/20=11
11=11