SOLUTION: Solve the system by substitution. 4y + 19 = x 3y - x = -13 Solve the system by elimination. 3x + 2y = 17 4x - 2y = 4

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Solve the system by substitution. 4y + 19 = x 3y - x = -13 Solve the system by elimination. 3x + 2y = 17 4x - 2y = 4      Log On


   

Question 906982: Solve the system by substitution.
4y + 19 = x
3y - x = -13
Solve the system by elimination.
3x + 2y = 17
4x - 2y = 4
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the system by substitution.
4y+%2B+19+=+x
3y+-+x+=+-13
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-x%2B4y+=+-19
-x%2B3y+=+-13
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Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
+system%28+%0D%0A++++-1%5Cx+%2B+4%5Cy+=+-19%2C%0D%0A++++-1%5Cx+%2B+3%5Cy+=+-13+%29%0D%0A++We'll use substitution. After moving 4*y to the right, we get:
-1%2Ax+=+-19+-+4%2Ay, or x+=+-19%2F-1+-+4%2Ay%2F-1. Substitute that
into another equation:
-1%2A%28-19%2F-1+-+4%2Ay%2F-1%29+%2B+3%5Cy+=+-13 and simplify: So, we know that y=-6. Since x+=+-19%2F-1+-+4%2Ay%2F-1, x=-5.

Answer: system%28+x=-5%2C+y=-6+%29.




Solve the system by elimination.
3x+%2B+2y+=+17+
4x+-+2y+=+4
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Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax%2B2%2Ay=17
4%2Ax-2%2Ay=4

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 3 and 4 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 4 is 12, we need to multiply both sides of the top equation by 4 and multiply both sides of the bottom equation by -3 like this:

4%2A%283%2Ax%2B2%2Ay%29=%2817%29%2A4 Multiply the top equation (both sides) by 4
-3%2A%284%2Ax-2%2Ay%29=%284%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
12%2Ax%2B8%2Ay=68
-12%2Ax%2B6%2Ay=-12

Notice how 12 and -12 add to zero (ie 12%2B-12=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2812%2Ax-12%2Ax%29%2B%288%2Ay%2B6%2Ay%29=68-12

%2812-12%29%2Ax%2B%288%2B6%29y=68-12

cross%2812%2B-12%29%2Ax%2B%288%2B6%29%2Ay=68-12 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

14%2Ay=56

y=56%2F14 Divide both sides by 14 to solve for y



y=4 Reduce


Now plug this answer into the top equation 3%2Ax%2B2%2Ay=17 to solve for x

3%2Ax%2B2%284%29=17 Plug in y=4


3%2Ax%2B8=17 Multiply



3%2Ax=17-8 Subtract 8 from both sides

3%2Ax=9 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%289%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=4

which also looks like

(3, 4)

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

3%2Ax%2B2%2Ay=17
4%2Ax-2%2Ay=4

we get



graph of 3%2Ax%2B2%2Ay=17 (red) 4%2Ax-2%2Ay=4 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (3,4). This verifies our answer.