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

Algebra ->  Customizable Word Problem Solvers  -> Misc -> 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       Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 121347: - 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
2x + 3y = 8 and 3x + 2y = 7
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

2%2Ax%2B3%2Ay=8
3%2Ax%2B2%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

3%2Ay=8-2%2AxSubtract 2%2Ax from both sides

y=%288-2%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=8%2F3-%282%2F3%29%2Ax Now we've fully isolated y

Since y equals 8%2F3-%282%2F3%29%2Ax we can substitute the expression 8%2F3-%282%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B2%2Ahighlight%28%288%2F3-%282%2F3%29%2Ax%29%29=7 Replace y with 8%2F3-%282%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B2%2A%288%2F3%29%2B2%28-2%2F3%29x=7 Distribute 2 to 8%2F3-%282%2F3%29%2Ax

3%2Ax%2B16%2F3-%284%2F3%29%2Ax=7 Multiply



3%2Ax%2B16%2F3-%284%2F3%29%2Ax=7 Reduce any fractions

3%2Ax-%284%2F3%29%2Ax=7-16%2F3 Subtract 16%2F3 from both sides


3%2Ax-%284%2F3%29%2Ax=21%2F3-16%2F3 Make 7 into a fraction with a denominator of 3


3%2Ax-%284%2F3%29%2Ax=5%2F3 Combine the terms on the right side



%289%2F3%29%2Ax-%284%2F3%29x=5%2F3 Make 3 into a fraction with a denominator of 3

%285%2F3%29%2Ax=5%2F3 Now combine the terms on the left side.


cross%28%283%2F5%29%285%2F3%29%29x=%285%2F3%29%283%2F5%29 Multiply both sides by 3%2F5. This will cancel out 5%2F3 and isolate x

So when we multiply 5%2F3 and 3%2F5 (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%281%29%2B2%2Ay=7 Plug in x=1 into the 2nd equation

3%2B2%2Ay=7 Multiply

2%2Ay=7-3Subtract 3 from both sides

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

cross%28%281%2F2%29%282%29%29%2Ay=%284%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=4%2F2 Multiply the terms on the right side


y=2 Reduce


So this is the other answer


y=2<---------------------------------Other answer


So our solution is

x=1 and y=2

which can also look like

(1,2)

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

2%2Ax%2B3%2Ay=8
3%2Ax%2B2%2Ay=7

we get


graph of 2%2Ax%2B3%2Ay=8 (red) and 3%2Ax%2B2%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 (1,2). This verifies our answer.


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

Plug in (1,2) into the system of equations


Let x=1 and y=2. Now plug those values into the equation 2%2Ax%2B3%2Ay=8

2%2A%281%29%2B3%2A%282%29=8 Plug in x=1 and y=2


2%2B6=8 Multiply


8=8 Add


8=8 Reduce. Since this equation is true the solution works.


So the solution (1,2) satisfies 2%2Ax%2B3%2Ay=8



Let x=1 and y=2. Now plug those values into the equation 3%2Ax%2B2%2Ay=7

3%2A%281%29%2B2%2A%282%29=7 Plug in x=1 and y=2


3%2B4=7 Multiply


7=7 Add


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


So the solution (1,2) satisfies 3%2Ax%2B2%2Ay=7


Since the solution (1,2) satisfies the system of equations


2%2Ax%2B3%2Ay=8
3%2Ax%2B2%2Ay=7


this verifies our answer.