SOLUTION: Find the point where the lines with the following two equations intersect: x + 2y = 3 and 4x - 3y = 1

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Question 777343: Find the point where the lines with the following two equations intersect: x + 2y = 3 and 4x - 3y = 1
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
x+%2B+2y+=+3 and
4x+-+3y+=+1
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Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B2%2Ay=3
4%2Ax-3%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

2%2Ay=3-1%2AxSubtract 1%2Ax from both sides

y=%283-1%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



y=3%2F2-%281%2F2%29%2Ax Now we've fully isolated y

Since y equals 3%2F2-%281%2F2%29%2Ax we can substitute the expression 3%2F2-%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B-3%2Ahighlight%28%283%2F2-%281%2F2%29%2Ax%29%29=1 Replace y with 3%2F2-%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax-3%2A%283%2F2%29-3%28-1%2F2%29x=1 Distribute -3 to 3%2F2-%281%2F2%29%2Ax

4%2Ax-9%2F2%2B%283%2F2%29%2Ax=1 Multiply



4%2Ax-9%2F2%2B%283%2F2%29%2Ax=1 Reduce any fractions

4%2Ax%2B%283%2F2%29%2Ax=1%2B9%2F2Add 9%2F2 to both sides


4%2Ax%2B%283%2F2%29%2Ax=2%2F2%2B9%2F2 Make 1 into a fraction with a denominator of 2


4%2Ax%2B%283%2F2%29%2Ax=11%2F2 Combine the terms on the right side



%288%2F2%29%2Ax%2B%283%2F2%29x=11%2F2 Make 4 into a fraction with a denominator of 2

%2811%2F2%29%2Ax=11%2F2 Now combine the terms on the left side.


cross%28%282%2F11%29%2811%2F2%29%29x=%2811%2F2%29%282%2F11%29 Multiply both sides by 2%2F11. This will cancel out 11%2F2 and isolate x

So when we multiply 11%2F2 and 2%2F11 (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

4%281%29-3%2Ay=1 Plug in x=1 into the 2nd equation

4-3%2Ay=1 Multiply

-3%2Ay=1-4Subtract 4 from both sides

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

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

y=-3%2F-3 Multiply the terms on the right side


y=1 Reduce


So this is the other answer


y=1<---------------------------------Other answer


So our solution is

x=1 and y=1

which can also look like

(1,1)

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

1%2Ax%2B2%2Ay=3
4%2Ax-3%2Ay=1

we get


graph of 1%2Ax%2B2%2Ay=3 (red) and 4%2Ax-3%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 (1,1). This verifies our answer.


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Check:

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


Let x=1 and y=1. Now plug those values into the equation 1%2Ax%2B2%2Ay=3

1%2A%281%29%2B2%2A%281%29=3 Plug in x=1 and y=1


1%2B2=3 Multiply


3=3 Add


3=3 Reduce. Since this equation is true the solution works.


So the solution (1,1) satisfies 1%2Ax%2B2%2Ay=3



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

4%2A%281%29-3%2A%281%29=1 Plug in x=1 and y=1


4-3=1 Multiply


1=1 Add


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


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


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


1%2Ax%2B2%2Ay=3
4%2Ax-3%2Ay=1


this verifies our answer.