SOLUTION: Could you please help me with this question? Solve: 1.5X – 2Y = 0 3X – 4Y = 7 Thank you.

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Question 90981: Could you please help me with this question?
Solve:
1.5X – 2Y = 0
3X – 4Y = 7
Thank you.
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

5%2Ax-2%2Ay=0
3%2Ax-4%2Ay=7

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=0-5%2AxSubtract 5%2Ax from both sides

y=%280-5%2Ax%29%2F-2 Divide both sides by -2.


Which breaks down and reduces to



y=0%2B%285%2F2%29%2Ax Now we've fully isolated y

Since y equals 0%2B%285%2F2%29%2Ax we can substitute the expression 0%2B%285%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-4%2Ahighlight%28%280%2B%285%2F2%29%2Ax%29%29=7 Replace y with 0%2B%285%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-4%2A%280%29-4%285%2F2%29x=7 Distribute -4 to 0%2B%285%2F2%29%2Ax

3%2Ax%2B0-%2820%2F2%29%2Ax=7 Multiply



3%2Ax%2B0-10%2Ax=7 Reduce any fractions

3%2Ax-10%2Ax=7%2B0Add 0 to both sides


3%2Ax-10%2Ax=7 Combine the terms on the right side



-7%2Ax=7 Now combine the terms on the left side.


cross%28%281%2F-7%29%28-7%2F1%29%29x=%287%2F1%29%281%2F-7%29 Multiply both sides by 1%2F-7. This will cancel out -7%2F1 and isolate x

So when we multiply 7%2F1 and 1%2F-7 (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

3%28-1%29-4%2Ay=7 Plug in x=-1 into the 2nd equation

-3-4%2Ay=7 Multiply

-4%2Ay=7%2B3Add 3 to both sides

-4%2Ay=10 Combine the terms on the right side

cross%28%281%2F-4%29%28-4%29%29%2Ay=%2810%2F1%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 on the left side.

y=10%2F-4 Multiply the terms on the right side


y=-5%2F2 Reduce


So this is the other answer


y=-5%2F2<---------------------------------Other answer


So our solution is

x=-1 and y=-5%2F2

which can also look like

(-1,-5%2F2)

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

5%2Ax-2%2Ay=0
3%2Ax-4%2Ay=7

we get


graph of 5%2Ax-2%2Ay=0 (red) and 3%2Ax-4%2Ay=7 (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,-5%2F2). This verifies our answer.


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

Plug in (-1,-5%2F2) into the system of equations


Let x=-1 and y=-5%2F2. Now plug those values into the equation 5%2Ax-2%2Ay=0

5%2A%28-1%29-2%2A%28-5%2F2%29=0 Plug in x=-1 and y=-5%2F2


-5%2B10%2F2=0 Multiply


0%2F2=0 Add


0=0 Reduce. Since this equation is true the solution works.


So the solution (-1,-5%2F2) satisfies 5%2Ax-2%2Ay=0



Let x=-1 and y=-5%2F2. Now plug those values into the equation 3%2Ax-4%2Ay=7

3%2A%28-1%29-4%2A%28-5%2F2%29=7 Plug in x=-1 and y=-5%2F2


-3%2B20%2F2=7 Multiply


14%2F2=7 Add


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


So the solution (-1,-5%2F2) satisfies 3%2Ax-4%2Ay=7


Since the solution (-1,-5%2F2) satisfies the system of equations


5%2Ax-2%2Ay=0
3%2Ax-4%2Ay=7


this verifies our answer.