SOLUTION: Graph 3x+4y=7 and 5x+2y=12. I know how to do this. The next part of the problem asks: Find the exact coordinates of their point of intersection, and check your answer by substituti

Question 123074: Graph 3x+4y=7 and 5x+2y=12. I know how to do this. The next part of the problem asks: Find the exact coordinates of their point of intersection, and check your answer by substitution in the given equations. How do you go about doing this ?
Found 2 solutions by jim_thompson5910, rapaljer:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Note: It will be very hard to find the exact coordinates of the intersection with just the graph (it turns out that the coordinates of the intersection are fractions). So to find the exact coordinates of the intersection, we need to use either substitution or elimination. So let's use substitution

Start with the given system of equations:

In order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.

So let's isolate y in the first equation

Start with the first equation

Subtract from both sides

Rearrange the equation

Divide both sides by

Break up the fraction

Reduce

---------------------

Since , we can now replace each in the second equation with to solve for

Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.

Distribute to

Multiply

Multiply both sides by the LCM of 4. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)

Distribute and multiply the LCM to each side

Combine like terms on the left side

Subtract 14 from both sides

Combine like terms on the right side

Divide both sides by 14 to isolate x

Reduce

-----------------First Answer------------------------------

So the first part of our answer is:

Since we know that we can plug it into the equation (remember we previously solved for in the first equation).

Start with the equation where was previously isolated.

Plug in

Multiply

Combine like terms and reduce. (note: if you need help with fractions, check out this solver)

-----------------Second Answer------------------------------

So the second part of our answer is:

-----------------Summary------------------------------

So our answers are:

and

which form the point

Now let's graph the two equations (if you need help with graphing, check out this solver)

From the graph, we can see that the two equations intersect at . This visually verifies our answer.

graph of (red) and (green) and the intersection of the lines (blue circle).

Check:

Start with the given system of equations:

Plug in and :

Multiply

Combine like terms

Since we have true equalities, this verifies our answer.
Answer by rapaljer(4671) (Show Source): You can put this solution on YOUR website!
3x+4y=7
5x+2y=12

You need to find a way to eliminate one of the variables, by multiplying one or both of the equations by some number(s), and adding them together. In this case, since you have a 4y and a 2y, if you want to make the 4y subtract out, you will need a -4y, so you should multiply the second equation by -2.

3x+4y=7
-10x-4y=-24

-7x = -17

Since this came out to such an ugly answer, I would eliminate the x terms in the same way. You have coefficients of x of 3 and 5. A "common number" would be 15, so multiply the first equation by 5 and the second equation by -3, which will give you a 15x and -15x, which subtract out.

3x+4y=7
5x+2y=12

5(3x+4y)=5(7)
-3(5x+2y)=-3(12)

15x+20y = 35
-15x-6y=-36

14y=-1
y=-1/14

This is REALLY ugly, so check it by substituting into the first equation:
3x+4y=7
3(17/7) +4(-1/14) = 7
51/7 -2/7 = 7
49/7=7

Some instructors require a check in BOTH equations. The second check will be similar to this!

R^2

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