SOLUTION: Please help! The problem is: Solve the system by using the substitute method. Determine whether the equations are independent, dependent or inconsistent. Eq 1 states: 3x

Algebra ->  Expressions-with-variables -> SOLUTION: Please help! The problem is: Solve the system by using the substitute method. Determine whether the equations are independent, dependent or inconsistent. Eq 1 states: 3x      Log On


   

Question 102664This question is from textbook
: Please help!
The problem is:
Solve the system by using the substitute method. Determine whether the equations are independent, dependent or inconsistent.
Eq 1 states: 3x + 5y = 25
Eq 2 states: x -2y = -10 This question is from textbook

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

3%2Ax%2B5%2Ay=25
1%2Ax-2%2Ay=-10

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=25-3%2AxSubtract 3%2Ax from both sides

y=%2825-3%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=5-%283%2F5%29%2Ax Now we've fully isolated y

Since y equals 5-%283%2F5%29%2Ax we can substitute the expression 5-%283%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-2%2Ahighlight%28%285-%283%2F5%29%2Ax%29%29=-10 Replace y with 5-%283%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-2%2A%285%29-2%28-3%2F5%29x=-10 Distribute -2 to 5-%283%2F5%29%2Ax

1%2Ax-10%2B%286%2F5%29%2Ax=-10 Multiply



1%2Ax-10%2B%286%2F5%29%2Ax=-10 Reduce any fractions

1%2Ax%2B%286%2F5%29%2Ax=-10%2B10Add 10 to both sides


1%2Ax%2B%286%2F5%29%2Ax=0 Combine the terms on the right side



%285%2F5%29%2Ax%2B%286%2F5%29x=0 Make 1 into a fraction with a denominator of 5

%2811%2F5%29%2Ax=0 Now combine the terms on the left side.


cross%28%285%2F11%29%2811%2F5%29%29x=%280%2F1%29%285%2F11%29 Multiply both sides by 5%2F11. This will cancel out 11%2F5 and isolate x

So when we multiply 0%2F1 and 5%2F11 (and simplify) we get



x=0 <---------------------------------One answer

Now that we know that x=0, lets substitute that in for x to solve for y

1%280%29-2%2Ay=-10 Plug in x=0 into the 2nd equation

0-2%2Ay=-10 Multiply

-2%2Ay=-10%2B0Add 0 to both sides

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

cross%28%281%2F-2%29%28-2%29%29%2Ay=%28-10%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

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


y=5 Reduce


So this is the other answer


y=5<---------------------------------Other answer


So our solution is

x=0 and y=5

which can also look like

(0,5)

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

3%2Ax%2B5%2Ay=25
1%2Ax-2%2Ay=-10

we get


graph of 3%2Ax%2B5%2Ay=25 (red) and 1%2Ax-2%2Ay=-10 (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 (0,5). This verifies our answer.


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Check:

Plug in (0,5) into the system of equations


Let x=0 and y=5. Now plug those values into the equation 3%2Ax%2B5%2Ay=25

3%2A%280%29%2B5%2A%285%29=25 Plug in x=0 and y=5


0%2B25=25 Multiply


25=25 Add


25=25 Reduce. Since this equation is true the solution works.


So the solution (0,5) satisfies 3%2Ax%2B5%2Ay=25



Let x=0 and y=5. Now plug those values into the equation 1%2Ax-2%2Ay=-10

1%2A%280%29-2%2A%285%29=-10 Plug in x=0 and y=5


0-10=-10 Multiply


-10=-10 Add


-10=-10 Reduce. Since this equation is true the solution works.


So the solution (0,5) satisfies 1%2Ax-2%2Ay=-10


Since the solution (0,5) satisfies the system of equations


3%2Ax%2B5%2Ay=25
1%2Ax-2%2Ay=-10


this verifies our answer.