SOLUTION: - Determine how many solutions exist - Use either elimination or substitution to find the solutions (if any) - Graph the two lines, labeling the x-intercepts, y-intercepts, and p

Algebra ->  Linear-equations -> SOLUTION: - Determine how many solutions exist - Use either elimination or substitution to find the solutions (if any) - Graph the two lines, labeling the x-intercepts, y-intercepts, and p      Log On


   

Question 142804: - Determine how many solutions exist
- Use either elimination or substitution to find the solutions (if any)
- Graph the two lines, labeling the x-intercepts, y-intercepts, and points
of intersection
x + y = 3 and y = x + 3
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

x+%2B+y+=+3 and y+=+x+%2B+3;
you can write them both in the standard form like this:
x+%2B+y+=+3
-x+%2B+y+=+3
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=3
-1%2Ax%2B1%2Ay=3

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

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

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


Which breaks down and reduces to



y=3-1%2Ax Now we've fully isolated y

Since y equals 3-1%2Ax we can substitute the expression 3-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-1%2Ax%2B1%2Ahighlight%28%283-1%2Ax%29%29=3 Replace y with 3-1%2Ax. Since this eliminates y, we can now solve for x.

-1%2Ax%2B1%2A%283%29%2B1%28-1%29x=3 Distribute 1 to 3-1%2Ax

-1%2Ax%2B3-1%2Ax=3 Multiply



-1%2Ax%2B3-1%2Ax=3 Reduce any fractions

-1%2Ax-1%2Ax=3-3 Subtract 3 from both sides


-1%2Ax-1%2Ax=0 Combine the terms on the right side



-2%2Ax=0 Now combine the terms on the left side.


cross%28%281%2F-2%29%28-2%2F1%29%29x=%280%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2%2F1 and isolate x

So when we multiply 0%2F1 and 1%2F-2 (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%2B1%2Ay=3 Plug in x=0 into the 2nd equation

0%2B1%2Ay=3 Multiply

1%2Ay=3%2B0Add 0 to both sides

1%2Ay=3 Combine the terms on the right side

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

y=3%2F1 Multiply the terms on the right side


y=3 Reduce


So this is the other answer


y=3<---------------------------------Other answer


So our solution is

x=0 and y=3

which can also look like

(0,3)

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

1%2Ax%2B1%2Ay=3
-1%2Ax%2B1%2Ay=3

we get


graph of 1%2Ax%2B1%2Ay=3 (red) and -1%2Ax%2B1%2Ay=3 (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 (0,3). This verifies our answer.


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Check:

Plug in (0,3) into the system of equations


Let x=0 and y=3. Now plug those values into the equation 1%2Ax%2B1%2Ay=3

1%2A%280%29%2B1%2A%283%29=3 Plug in x=0 and y=3


0%2B3=3 Multiply


3=3 Add


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


So the solution (0,3) satisfies 1%2Ax%2B1%2Ay=3



Let x=0 and y=3. Now plug those values into the equation -1%2Ax%2B1%2Ay=3

-1%2A%280%29%2B1%2A%283%29=3 Plug in x=0 and y=3


0%2B3=3 Multiply


3=3 Add


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


So the solution (0,3) satisfies -1%2Ax%2B1%2Ay=3


Since the solution (0,3) satisfies the system of equations


1%2Ax%2B1%2Ay=3
-1%2Ax%2B1%2Ay=3


this verifies our answer.