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Systems of Equations

There are many strategies we can use to solve a system of equations, such as elimination or substitution.

  • In elimination, we subtract or add the equations entirely to cancel out a variable.
  • In substitution, we substitute one equation into the other.

Solution

[tex]y=3x+6\\y=(x+4)^2-10[/tex]

Since y has already be isolated in both equations, we can just use substitution to equate them:

[tex]3x+6=(x+4)^2-10[/tex]

Now, combine like terms:

[tex]3x=(x+4)^2-16[/tex]

Expand the binomial:

[tex]3x=x^2+8x+16-16\\3x=x^2+8x\\0=x^2+5x[/tex]

Find the zeroes:

[tex]0=x(x+5)[/tex]

Therefore, x can be 0 or -5.

Now, we can plug these values of x back into one of the equations to determine the y coordinate:

[tex]y=3x+6\\y=3(0)+6\\y=6[/tex]

Therefore, one solution is (0,6).

[tex]y=3x+6\\y=3(-5)+6\\y=-15+6\\y=-9[/tex]

Another solutions is (-5,-9).

Answer

(0,6), (-5,-9)

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