How do you solve the system of equations 3x + y = 233x+y=23 and 4x - y = 194xy=19?

2 Answers
Jan 7, 2018

X=6, Y=5

Explanation:

Use simultaneous equations (multiplying both equations, and eliminating either x or y by adding or subtracting to find the value of one, then plugging the value back in to find the other value)

The lowest common multiple of 3 and 4 is 12
3*4=1234=12 so we have to multiply the first equation by 4
4*3=1243=12 so we have to multiply the second equation by 3

This is the result:
12x+4y=9212x+4y=92
12x-3y=5712x3y=57

From here as x is the same, you can subtract the bottom equation from the top leaving you with y. After doing this the result is:

7y=35 7y=35
y=35/7=5 y=357=5

Then we can plug this value back into the equations, then we can solve to find x.
3x+y=233x+y=23
3x+5=233x+5=23
3x=183x=18 (-5)
x=18/3=6x=183=6

We have got x=6x=6 and y=5y=5, but we have to check if these values work for both equations so lets check.

4x-y=194xy=19
4*6-5=19465=19
24-5=19245=19

So therefore these values fit both equations and we are correct.

Jan 7, 2018

The point of intersection is (6,5)(6,5)

Explanation:

You can also use substitution. The resulting values for xx and yy are the point of intersection of the two equations.

Equation 1: 3x+y=233x+y=23

Equation 2: 4x-y=194xy=19

Solve Equation 1 for yy.

3x+y=233x+y=23

Subtract 3x3x from both sides.

y=23-3xy=233x

Substitute 23-3x233x for yy in Equation 2 and solve for xx.

4x-y=194xy=19

4x-(23-3x)=194x(233x)=19

Simplify.

4x-23+3x=194x23+3x=19

Add 2323 to both sides.

4x+3x=19+234x+3x=19+23

Simplify.

7x=427x=42

Divide both sides by 77.

x=42/7x=427

x=6x=6

Substitute 66 for xx into Equation 1 and solve for yy.

3x+y=233x+y=23

3(6)+y=233(6)+y=23

Simplify.

18+y=2318+y=23

Subtract 1818 from both sides.

y=23-18y=2318

y=5y=5

The point of intersection is (6,5)(6,5).

Equation 1:

3(6)+5=233(6)+5=23

18+5=2318+5=23

23=2323=23 sqrt

Equation 2:

4(6)-5=194(6)5=19

24-5=19245=19

19=1919=19 sqrt

graph{(y+3x-23)(-y+4x-19)=0 [-15.69, 16.34, -5.83, 10.19]}